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Since theory suggests that dark energy does not cluster in the usual way it is the best explanation for the "missing" energy density. Dark energy also helps to explain two geometrical measures of the overall curvature of the universe, one using the frequency of gravitational lenses, and the other using the characteristic pattern of the large-scale structure--baryon acoustic oscillations--as a cosmic ruler.
Negative pressure is believed to be a property of vacuum energy, but the exact nature and existence of dark energy remains one of the great mysteries of the Big Bang. Results from the WMAP team in 2008 are in accordance with a universe that consists of 73% dark energy, 23% dark matter, 4.6% regular matter and less than 1% neutrinos. According to theory, the energy density in matter decreases with the expansion of the universe, but the dark energy density remains constant (or nearly so) as the universe expands. Therefore, matter made up a larger fraction of the total energy of the universe in the past than it does today, but its fractional contribution will fall in the far future as dark energy becomes even more dominant.
The dark energy component of the universe has been explained by theorists using a variety of competing theories including Einstein's cosmological constant but also extending to more exotic forms of quintessence or other modified gravity schemes.
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Therefore, matter made up a larger fraction of the total energy of the universe in the past than it does today, but its fractional contribution will fall in the far future as dark energy becomes even more dominant.
The dark energy component of the universe has been explained by theorists using a variety of competing theories including Einstein's cosmological constant but also extending to more exotic forms of quintessence or other modified gravity schemes. A cosmological constant problem, sometimes called the "most embarrassing problem in physics", results from the apparent discrepancy between the measured energy density of dark energy, and the one naively predicted from Planck units.
Dark matter
During the 1970s and the 1980s, various observations showed that there is not sufficient visible matter in the universe to account for the apparent strength of gravitational forces within and between galaxies. This led to the idea that up to 90% of the matter in the universe is dark matter that does not emit light or interact with normal baryonic matter. In addition, the assumption that the universe is mostly normal matter led to predictions that were strongly inconsistent with observations. In particular, the universe today is far more lumpy and contains far less deuterium than can be accounted for without dark matter.
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In addition, the assumption that the universe is mostly normal matter led to predictions that were strongly inconsistent with observations. In particular, the universe today is far more lumpy and contains far less deuterium than can be accounted for without dark matter. While dark matter has always been controversial, it is inferred by various observations: the anisotropies in the CMB, the galaxy rotation problem, galaxy cluster velocity dispersions, large-scale structure distributions, gravitational lensing studies, and X-ray measurements of galaxy clusters.
Indirect evidence for dark matter comes from its gravitational influence on other matter, as no dark matter particles have been observed in laboratories. Many particle physics candidates for dark matter have been proposed, and several projects to detect them directly are underway.
Additionally, there are outstanding problems associated with the currently favored cold dark matter model which include the dwarf galaxy problem and the cuspy halo problem. Alternative theories have been proposed that do not require a large amount of undetected matter, but instead modify the laws of gravity established by Newton and Einstein; yet no alternative theory has been as successful as the cold dark matter proposal in explaining all extant observations.
### Horizon problem
The horizon problem results from the premise that information cannot travel faster than light.
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Alternative theories have been proposed that do not require a large amount of undetected matter, but instead modify the laws of gravity established by Newton and Einstein; yet no alternative theory has been as successful as the cold dark matter proposal in explaining all extant observations.
### Horizon problem
The horizon problem results from the premise that information cannot travel faster than light. In a universe of finite age this sets a limit—the particle horizon—on the separation of any two regions of space that are in causal contact. The observed isotropy of the CMB is problematic in this regard: if the universe had been dominated by radiation or matter at all times up to the epoch of last scattering, the particle horizon at that time would correspond to about 2 degrees on the sky. There would then be no mechanism to cause wider regions to have the same temperature.
A resolution to this apparent inconsistency is offered by inflation theory in which a homogeneous and isotropic scalar energy field dominates the universe at some very early period (before baryogenesis). During inflation, the universe undergoes exponential expansion, and the particle horizon expands much more rapidly than previously assumed, so that regions presently on opposite sides of the observable universe are well inside each other's particle horizon.
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A resolution to this apparent inconsistency is offered by inflation theory in which a homogeneous and isotropic scalar energy field dominates the universe at some very early period (before baryogenesis). During inflation, the universe undergoes exponential expansion, and the particle horizon expands much more rapidly than previously assumed, so that regions presently on opposite sides of the observable universe are well inside each other's particle horizon. The observed isotropy of the CMB then follows from the fact that this larger region was in causal contact before the beginning of inflation.
Heisenberg's uncertainty principle predicts that during the inflationary phase there would be quantum thermal fluctuations, which would be magnified to a cosmic scale. These fluctuations served as the seeds for all the current structures in the universe. Inflation predicts that the primordial fluctuations are nearly scale invariant and Gaussian, which has been confirmed by measurements of the CMB.
A related issue to the classic horizon problem arises because in most standard cosmological inflation models, inflation ceases well before electroweak symmetry breaking occurs, so inflation should not be able to prevent large-scale discontinuities in the electroweak vacuum since distant parts of the observable universe were causally separate when the electroweak epoch ended.
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Inflation predicts that the primordial fluctuations are nearly scale invariant and Gaussian, which has been confirmed by measurements of the CMB.
A related issue to the classic horizon problem arises because in most standard cosmological inflation models, inflation ceases well before electroweak symmetry breaking occurs, so inflation should not be able to prevent large-scale discontinuities in the electroweak vacuum since distant parts of the observable universe were causally separate when the electroweak epoch ended.
### Magnetic monopoles
The magnetic monopole objection was raised in the late 1970s. Grand unified theories (GUTs) predicted topological defects in space that would manifest as magnetic monopoles. These objects would be produced efficiently in the hot early universe, resulting in a density much higher than is consistent with observations, given that no monopoles have been found. This problem is resolved by cosmic inflation, which removes all point defects from the observable universe, in the same way that it drives the geometry to flatness.
### Flatness problem
The flatness problem (also known as the oldness problem) is an observational problem associated with a FLRW. The universe may have positive, negative, or zero spatial curvature depending on its total energy density.
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### Flatness problem
The flatness problem (also known as the oldness problem) is an observational problem associated with a FLRW. The universe may have positive, negative, or zero spatial curvature depending on its total energy density. Curvature is negative if its density is less than the critical density; positive if greater; and zero at the critical density, in which case space is said to be flat. Observations indicate the universe is consistent with being flat.
The problem is that any small departure from the critical density grows with time, and yet the universe today remains very close to flat. Given that a natural timescale for departure from flatness might be the Planck time, 10−43 seconds, the fact that the universe has reached neither a heat death nor a Big Crunch after billions of years requires an explanation. For instance, even at the relatively late age of a few minutes (the time of nucleosynthesis), the density of the universe must have been within one part in 1014 of its critical value, or it would not exist as it does today.
## Misconceptions
One of the common misconceptions about the Big Bang model is that it fully explains the origin of the universe. However, the Big Bang model does not describe how energy, time, and space were caused, but rather it describes the emergence of the present universe from an ultra-dense and high-temperature initial state.
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## Misconceptions
One of the common misconceptions about the Big Bang model is that it fully explains the origin of the universe. However, the Big Bang model does not describe how energy, time, and space were caused, but rather it describes the emergence of the present universe from an ultra-dense and high-temperature initial state. It is misleading to visualize the Big Bang by comparing its size to everyday objects. When the size of the universe at Big Bang is described, it refers to the size of the observable universe, and not the entire universe.
Another common misconception is that the Big Bang must be understood as the expansion of space and not in terms of the contents of space exploding apart. In fact, either description can be accurate. The expansion of space (implied by the FLRW metric) is only a mathematical convention, corresponding to a choice of coordinates on spacetime. There is no generally covariant sense in which space expands.
The recession speeds associated with Hubble's law are not velocities in a relativistic sense (for example, they are not related to the spatial components of 4-velocities). Therefore, it is not remarkable that according to Hubble's law, galaxies farther than the Hubble distance recede faster than the speed of light. Such recession speeds do not correspond to faster-than-light travel.
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Therefore, it is not remarkable that according to Hubble's law, galaxies farther than the Hubble distance recede faster than the speed of light. Such recession speeds do not correspond to faster-than-light travel.
Many popular accounts attribute the cosmological redshift to the expansion of space. This can be misleading because the expansion of space is only a coordinate choice. The most natural interpretation of the cosmological redshift is that it is a Doppler shift.
## Implications
Given current understanding, scientific extrapolations about the future of the universe are only possible for finite durations, albeit for much longer periods than the current age of the universe. Anything beyond that becomes increasingly speculative. Likewise, at present, a proper understanding of the origin of the universe can only be subject to conjecture.
### Pre–Big Bang cosmology
The Big Bang explains the evolution of the universe from a starting density and temperature that is well beyond humanity's capability to replicate, so extrapolations to the most extreme conditions and earliest times are necessarily more speculative. Lemaître called this initial state the "primeval atom" while Gamow called the material "ylem". How the initial state of the universe originated is still an open question, but the Big Bang model does constrain some of its characteristics.
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Lemaître called this initial state the "primeval atom" while Gamow called the material "ylem". How the initial state of the universe originated is still an open question, but the Big Bang model does constrain some of its characteristics. For example, if specific laws of nature were to come to existence in a random way, inflation models show, some combinations of these are far more probable, partly explaining why our Universe is rather stable. Another possible explanation for the stability of the Universe could be a hypothetical multiverse, which assumes every possible universe to exist, and thinking species could only emerge in those stable enough. A flat universe implies a balance between gravitational potential energy and other energy forms, requiring no additional energy to be created.
The Big Bang theory is built upon the equations of classical general relativity, which are not expected to be valid at the origin of cosmic time, as the temperature of the universe approaches the Planck scale. Correcting this will require the development of a correct treatment of quantum gravity. Certain quantum gravity treatments, such as the Wheeler–DeWitt equation, imply that time itself could be an emergent property. As such, physics may conclude that time did not exist before the Big Bang.
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Certain quantum gravity treatments, such as the Wheeler–DeWitt equation, imply that time itself could be an emergent property. As such, physics may conclude that time did not exist before the Big Bang.
While it is not known what could have preceded the hot dense state of the early universe or how and why it originated, or even whether such questions are sensible, speculation abounds on the subject of "cosmogony".
Some speculative proposals in this regard, each of which entails untested hypotheses, are:
- The simplest models, in which the Big Bang was caused by quantum fluctuations. That scenario had very little chance of happening, but, according to the totalitarian principle, even the most improbable event will eventually happen. It took place instantly, in our perspective, due to the absence of perceived time before the Big Bang.
- Emergent Universe models, which feature a low-activity past-eternal era before the Big Bang, resembling ancient ideas of a cosmic egg and birth of the world out of primordial chaos.
- Models in which the whole of spacetime is finite, including the Hartle–Hawking no-boundary condition. For these cases, the Big Bang does represent the limit of time but without a singularity.
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It took place instantly, in our perspective, due to the absence of perceived time before the Big Bang.
- Emergent Universe models, which feature a low-activity past-eternal era before the Big Bang, resembling ancient ideas of a cosmic egg and birth of the world out of primordial chaos.
- Models in which the whole of spacetime is finite, including the Hartle–Hawking no-boundary condition. For these cases, the Big Bang does represent the limit of time but without a singularity. In such a case, the universe is self-sufficient.
- Brane cosmology models, in which inflation is due to the movement of branes in string theory; the pre-Big Bang model; the ekpyrotic model, in which the Big Bang is the result of a collision between branes; and the cyclic model, a variant of the ekpyrotic model in which collisions occur periodically. In the latter model the Big Bang was preceded by a Big Crunch and the universe cycles from one process to the other.
- Eternal inflation, in which universal inflation ends locally here and there in a random fashion, each end-point leading to a bubble universe, expanding from its own big bang. This is sometimes referred to as pre-big bang inflation.
Proposals in the last two categories see the Big Bang as an event in either a much larger and older universe or in a multiverse.
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This is sometimes referred to as pre-big bang inflation.
Proposals in the last two categories see the Big Bang as an event in either a much larger and older universe or in a multiverse.
### Ultimate fate of the universe
Before observations of dark energy, cosmologists considered two scenarios for the future of the universe. If the mass density of the universe were greater than the critical density, then the universe would reach a maximum size and then begin to collapse. It would become denser and hotter again, ending with a state similar to that in which it started—a Big Crunch.
Alternatively, if the density in the universe were equal to or below the critical density, the expansion would slow down but never stop. Star formation would cease with the consumption of interstellar gas in each galaxy; stars would burn out, leaving white dwarfs, neutron stars, and black holes. Collisions between these would result in mass accumulating into larger and larger black holes. The average temperature of the universe would very gradually asymptotically approach absolute zero—a Big Freeze. Moreover, if protons are unstable, then baryonic matter would disappear, leaving only radiation and black holes. Eventually, black holes would evaporate by emitting Hawking radiation.
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Moreover, if protons are unstable, then baryonic matter would disappear, leaving only radiation and black holes. Eventually, black holes would evaporate by emitting Hawking radiation. The entropy of the universe would increase to the point where no organized form of energy could be extracted from it, a scenario known as heat death.
Modern observations of accelerating expansion imply that more and more of the currently visible universe will pass beyond our event horizon and out of contact with us. The eventual result is not known. The ΛCDM model of the universe contains dark energy in the form of a cosmological constant. This theory suggests that only gravitationally bound systems, such as galaxies, will remain together, and they too will be subject to heat death as the universe expands and cools. Other explanations of dark energy, called phantom energy theories, suggest that ultimately galaxy clusters, stars, planets, atoms, nuclei, and matter itself will be torn apart by the ever-increasing expansion in a so-called Big Rip.
### Religious and philosophical interpretations
As a description of the origin of the universe, the Big Bang has significant bearing on religion and philosophy. As a result, it has become one of the liveliest areas in the discourse between science and religion.
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A Fourier series () is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in .
The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. The figures below illustrate some partial Fourier series results for the components of a square wave.
Fourier series are closely related to the Fourier transform, a more general tool that can even find the frequency information for functions that are not periodic.
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The figures below illustrate some partial Fourier series results for the components of a square wave.
Fourier series are closely related to the Fourier transform, a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions can be identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by
$$
\mathbb{T}
$$
or
$$
S_1
$$
. The Fourier transform is also part of Fourier analysis, but is defined for functions on
$$
\mathbb{R}^n
$$
.
Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. Many other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis.
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Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. Many other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis.
## History
The Fourier series is named in honor of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies), and publishing his Théorie analytique de la chaleur (Analytical theory of heat) in 1822. The Mémoire introduced Fourier analysis, specifically Fourier series. Through Fourier's research the fact was established that an arbitrary (at first, continuous and later generalized to any piecewise-smooth) function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the French Academy.
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Through Fourier's research the fact was established that an arbitrary (at first, continuous and later generalized to any piecewise-smooth) function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the French Academy. Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles.
Independently of Fourier, astronomer Friedrich Wilhelm Bessel introduced Fourier series to solve Kepler's equation. His work was published in 1819, unaware of Fourier's work which remained unpublished until 1822.
The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series.
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Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series.
From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet and Bernhard Riemann expressed Fourier's results with greater precision and formality.
Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids. The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, shell theory, etc.
### Beginnings
Joseph Fourier wrote
This immediately gives any coefficient ak of the trigonometric series for φ(y) for any function which has such an expansion. It works because if φ has such an expansion, then (under suitable convergence assumptions)
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### Beginnings
Joseph Fourier wrote
This immediately gives any coefficient ak of the trigonometric series for φ(y) for any function which has such an expansion. It works because if φ has such an expansion, then (under suitable convergence assumptions) the integral
$$
\begin{align}
&\int_{-1}^1\varphi(y)\cos(2k+1)\frac{\pi y}{2}\,dy \\
&= \int_{-1}^1\left(a\cos\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+\cdots\right)\,dy
\end{align}
$$
can be carried out term-by-term. But all terms involving
$$
\cos(2j+1)\frac{\pi y}{2} \cos(2k+1)\frac{\pi y}{2}
$$
for vanish when integrated from −1 to 1, leaving only the
$$
k^{\text{th}}
$$
term, which is 1.
In these few lines, which are close to the modern formalism used in Fourier series, Fourier revolutionized both mathematics and physics.
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the integral
$$
\begin{align}
&\int_{-1}^1\varphi(y)\cos(2k+1)\frac{\pi y}{2}\,dy \\
&= \int_{-1}^1\left(a\cos\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+\cdots\right)\,dy
\end{align}
$$
can be carried out term-by-term. But all terms involving
$$
\cos(2j+1)\frac{\pi y}{2} \cos(2k+1)\frac{\pi y}{2}
$$
for vanish when integrated from −1 to 1, leaving only the
$$
k^{\text{th}}
$$
term, which is 1.
In these few lines, which are close to the modern formalism used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and Gauss, Fourier believed that such trigonometric series could represent any arbitrary function.
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But all terms involving
$$
\cos(2j+1)\frac{\pi y}{2} \cos(2k+1)\frac{\pi y}{2}
$$
for vanish when integrated from −1 to 1, leaving only the
$$
k^{\text{th}}
$$
term, which is 1.
In these few lines, which are close to the modern formalism used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and Gauss, Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis.
When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.
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In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis.
When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.
### Fourier's motivation
The Fourier series expansion of the sawtooth function (below) looks more complicated than the simple formula
$$
s(x)=\tfrac{x}{\pi}
$$
, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the heat equation. For example, consider a metal plate in the shape of a square whose sides measure
$$
\pi
$$
meters, with coordinates
$$
(x,y) \in [0,\pi] \times [0,\pi]
$$
.
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While there are many applications, Fourier's motivation was in solving the heat equation. For example, consider a metal plate in the shape of a square whose sides measure
$$
\pi
$$
meters, with coordinates
$$
(x,y) \in [0,\pi] \times [0,\pi]
$$
. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by
$$
y=\pi
$$
, is maintained at the temperature gradient
$$
T(x,\pi)=x
$$
degrees Celsius, for
$$
x
$$
in
$$
(0,\pi)
$$
, then one can show that the stationary heat distribution (or the heat distribution after a long time has elapsed) is given by
$$
T(x,y) = 2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx) {\sinh(ny) \over \sinh(n\pi)}.
$$
Here, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of the equation from
### Analysis
§
#### Example
by
$$
\sinh(ny)/\sinh(n\pi)
$$
.
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### Analysis
§
#### Example
by
$$
\sinh(ny)/\sinh(n\pi)
$$
. While our example function
$$
s(x)
$$
seems to have a needlessly complicated Fourier series, the heat distribution
$$
T(x,y)
$$
is nontrivial. The function
$$
T
$$
cannot be written as a closed-form expression. This method of solving the heat problem was made possible by Fourier's work.
### Other applications
Another application is to solve the Basel problem by using
### Parseval's theorem
. The example generalizes and one may compute ζ(2n), for any positive integer n.
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### Parseval's theorem
. The example generalizes and one may compute ζ(2n), for any positive integer n.
## Definition
The Fourier series of a complex-valued -periodic function
$$
s(x)
$$
, integrable over the interval
$$
[0,P]
$$
on the real line, is defined as a trigonometric series of the form
$$
\sum_{n=-\infty}^\infty c_n e^{i 2\pi \tfrac{n}{P} x },
$$
such that the Fourier coefficients
$$
c_n
$$
are complex numbers defined by the integral
$$
c_n = \frac{1}{P}\int_0^P s(x)\ e^{-i 2\pi \tfrac{n}{P} x }\,dx.
$$
The series does not necessarily converge (in the pointwise sense) and, even if it does, it is not necessarily equal to
$$
s(x)
$$
.
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The example generalizes and one may compute ζ(2n), for any positive integer n.
## Definition
The Fourier series of a complex-valued -periodic function
$$
s(x)
$$
, integrable over the interval
$$
[0,P]
$$
on the real line, is defined as a trigonometric series of the form
$$
\sum_{n=-\infty}^\infty c_n e^{i 2\pi \tfrac{n}{P} x },
$$
such that the Fourier coefficients
$$
c_n
$$
are complex numbers defined by the integral
$$
c_n = \frac{1}{P}\int_0^P s(x)\ e^{-i 2\pi \tfrac{n}{P} x }\,dx.
$$
The series does not necessarily converge (in the pointwise sense) and, even if it does, it is not necessarily equal to
$$
s(x)
$$
. Only when certain conditions are satisfied (e.g. if
$$
s(x)
$$
is continuously differentiable) does the Fourier series converge to
$$
s(x)
$$
, i.e.,
$$
s(x) = \sum_{n=-\infty}^\infty c_n e^{i 2\pi \tfrac{n}{P} x }.
$$
For functions satisfying the Dirichlet sufficiency conditions, pointwise convergence holds.
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## Definition
The Fourier series of a complex-valued -periodic function
$$
s(x)
$$
, integrable over the interval
$$
[0,P]
$$
on the real line, is defined as a trigonometric series of the form
$$
\sum_{n=-\infty}^\infty c_n e^{i 2\pi \tfrac{n}{P} x },
$$
such that the Fourier coefficients
$$
c_n
$$
are complex numbers defined by the integral
$$
c_n = \frac{1}{P}\int_0^P s(x)\ e^{-i 2\pi \tfrac{n}{P} x }\,dx.
$$
The series does not necessarily converge (in the pointwise sense) and, even if it does, it is not necessarily equal to
$$
s(x)
$$
. Only when certain conditions are satisfied (e.g. if
$$
s(x)
$$
is continuously differentiable) does the Fourier series converge to
$$
s(x)
$$
, i.e.,
$$
s(x) = \sum_{n=-\infty}^\infty c_n e^{i 2\pi \tfrac{n}{P} x }.
$$
For functions satisfying the Dirichlet sufficiency conditions, pointwise convergence holds. However, these are not necessary conditions and there are many theorems about different types of convergence of Fourier series (e.g. uniform convergence or mean convergence).
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Only when certain conditions are satisfied (e.g. if
$$
s(x)
$$
is continuously differentiable) does the Fourier series converge to
$$
s(x)
$$
, i.e.,
$$
s(x) = \sum_{n=-\infty}^\infty c_n e^{i 2\pi \tfrac{n}{P} x }.
$$
For functions satisfying the Dirichlet sufficiency conditions, pointwise convergence holds. However, these are not necessary conditions and there are many theorems about different types of convergence of Fourier series (e.g. uniform convergence or mean convergence). The definition naturally extends to the Fourier series of a (periodic) distribution
$$
s
$$
(also called Fourier-Schwartz series). Then the Fourier series converges to
$$
s(x)
$$
in the distribution sense.
The process of determining the Fourier coefficients of a given function or signal is called analysis, while forming the associated trigonometric series (or its various approximations) is called synthesis.
### Synthesis
A Fourier series can be written in several equivalent forms, shown here as the
$$
N^\text{th}
$$
partial sums
$$
s_N(x)
$$
of the Fourier series of
$$
s(x)
$$
:
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The process of determining the Fourier coefficients of a given function or signal is called analysis, while forming the associated trigonometric series (or its various approximations) is called synthesis.
### Synthesis
A Fourier series can be written in several equivalent forms, shown here as the
$$
N^\text{th}
$$
partial sums
$$
s_N(x)
$$
of the Fourier series of
$$
s(x)
$$
:
The harmonics are indexed by an integer,
$$
n,
$$
which is also the number of cycles the corresponding sinusoids make in interval
$$
P
$$
. Therefore, the sinusoids have:
- a wavelength equal to
$$
\tfrac{P}{n}
$$
in the same units as
$$
x
$$
.
- a frequency equal to
$$
\tfrac{n}{P}
$$
in the reciprocal units of
$$
x
$$
.
These series can represent functions that are just a sum of one or more frequencies in the harmonic spectrum. In the limit
$$
N\to\infty
$$
, a trigonometric series can also represent the intermediate frequencies or non-sinusoidal functions because of the infinite number of terms.
Analysis
The coefficients can be given/assumed, such as a music synthesizer or time samples of a waveform.
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In the limit
$$
N\to\infty
$$
, a trigonometric series can also represent the intermediate frequencies or non-sinusoidal functions because of the infinite number of terms.
Analysis
The coefficients can be given/assumed, such as a music synthesizer or time samples of a waveform. In the latter case, the exponential form of Fourier series synthesizes a discrete-time Fourier transform where variable
$$
x
$$
represents frequency instead of time. In general, the coefficients are determined by analysis of a given function
$$
s(x)
$$
whose domain of definition is an interval of length
$$
P
$$
.
The
$$
\tfrac{2}{P}
$$
scale factor follows from substituting into and utilizing the orthogonality of the trigonometric system. The equivalence of and follows from Euler's formula
$$
\cos x = \frac{e^{ix} + e^{-ix}}{2}, \quad \sin x = \frac{e^{ix} - e^{-ix}}{2i},
$$
resulting in:
with
$$
c_{0}
$$
being the mean value of
$$
s
$$
on the interval
$$
P
$$
.
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Conversely:
Example
Consider a sawtooth function:
$$
s(x) = s(x + 2\pi k) = \frac{x}{\pi}, \quad \mathrm{for } -\pi < x < \pi,\text{ and } k \in \mathbb{Z}.
$$
In this case, the Fourier coefficients are given by
$$
\begin{align}
a_0 &= 0.\\
a_n & = \frac{1}{\pi}\int_{-\pi}^{\pi}s(x) \cos(nx)\,dx = 0, \quad n \ge 1. \\
b_n & = \frac{1}{\pi}\int_{-\pi}^{\pi}s(x) \sin(nx)\, dx\\
&= -\frac{2}{\pi n}\cos(n\pi) + \frac{2}{\pi^2 n^2}\sin(n\pi)\\
&= \frac{2\,(-1)^{n+1}}{\pi n}, \quad n \ge 1.\end{align}
$$
It can be shown that the Fourier series converges to
$$
s(x)
$$
at every point
$$
x
$$
where
$$
s
$$
is differentiable, and therefore:
$$
\begin{align}
s(x) &= a_0 + \sum_{n=1}^\infty \left[a_n\cos\left(nx\right)+b_n sin\left(nx\right)\right] \\[4pt]
&=\frac{2}{\pi}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx), \quad \mathrm{for}\ (x-\pi)\ \text{is not a multiple of}\ 2\pi.
\end{align}
$$
When
$$
x=\pi
$$
, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of
$$
s
$$
at
$$
x=\pi
$$
.
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This is a particular instance of the Dirichlet theorem for Fourier series.
This example leads to a solution of the Basel problem.
### Amplitude-phase form
If the function
$$
s(x)
$$
is real-valued then the Fourier series can also be represented as
where _ BLOCK1_ is the amplitude and
$$
\varphi_{n}
$$
is the phase shift of the
$$
n^{th}
$$
harmonic.
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### Amplitude-phase form
If the function
$$
s(x)
$$
is real-valued then the Fourier series can also be represented as
where _ BLOCK1_ is the amplitude and
$$
\varphi_{n}
$$
is the phase shift of the
$$
n^{th}
$$
harmonic.
The equivalence of and follows from the trigonometric identity:
$$
\cos\left(2\pi \tfrac{n}{P}x-\varphi_n\right) = \cos(\varphi_n)\cos\left(2\pi \tfrac{n}{P} x\right)
+ \sin(\varphi_n)\sin\left(2\pi \tfrac{n}{P} x\right),
$$
which implies
$$
a_n = A_n \cos(\varphi_n)\quad \text{and}\quad b_n = A_n \sin(\varphi_n)
$$
are the rectangular coordinates of a vector with polar coordinates
$$
A_n
$$
and
$$
\varphi_n
$$
given by
$$
A_n = \sqrt{a_n^2 + b_n^2}\quad \text{and}\quad \varphi_n = \operatorname{Arg}(c_n) = \operatorname{atan2}(b_n, a_n)
$$
where _
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BLOCK1_ is the amplitude and
$$
\varphi_{n}
$$
is the phase shift of the
$$
n^{th}
$$
harmonic.
The equivalence of and follows from the trigonometric identity:
$$
\cos\left(2\pi \tfrac{n}{P}x-\varphi_n\right) = \cos(\varphi_n)\cos\left(2\pi \tfrac{n}{P} x\right)
+ \sin(\varphi_n)\sin\left(2\pi \tfrac{n}{P} x\right),
$$
which implies
$$
a_n = A_n \cos(\varphi_n)\quad \text{and}\quad b_n = A_n \sin(\varphi_n)
$$
are the rectangular coordinates of a vector with polar coordinates
$$
A_n
$$
and
$$
\varphi_n
$$
given by
$$
A_n = \sqrt{a_n^2 + b_n^2}\quad \text{and}\quad \varphi_n = \operatorname{Arg}(c_n) = \operatorname{atan2}(b_n, a_n)
$$
where _ BLOCK9_ is the argument of
$$
c_{n}
$$
.
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The equivalence of and follows from the trigonometric identity:
$$
\cos\left(2\pi \tfrac{n}{P}x-\varphi_n\right) = \cos(\varphi_n)\cos\left(2\pi \tfrac{n}{P} x\right)
+ \sin(\varphi_n)\sin\left(2\pi \tfrac{n}{P} x\right),
$$
which implies
$$
a_n = A_n \cos(\varphi_n)\quad \text{and}\quad b_n = A_n \sin(\varphi_n)
$$
are the rectangular coordinates of a vector with polar coordinates
$$
A_n
$$
and
$$
\varphi_n
$$
given by
$$
A_n = \sqrt{a_n^2 + b_n^2}\quad \text{and}\quad \varphi_n = \operatorname{Arg}(c_n) = \operatorname{atan2}(b_n, a_n)
$$
where _ BLOCK9_ is the argument of
$$
c_{n}
$$
.
An example of determining the parameter
$$
\varphi_n
$$
for one value of
$$
n
$$
is shown in Figure 2.
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BLOCK9_ is the argument of
$$
c_{n}
$$
.
An example of determining the parameter
$$
\varphi_n
$$
for one value of
$$
n
$$
is shown in Figure 2. It is the value of
$$
\varphi
$$
at the maximum correlation between
$$
s(x)
$$
and a cosine template,
$$
\cos(2\pi \tfrac{n}{P} x - \varphi)
$$
.
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The blue graph is the cross-correlation function, also known as a matched filter:
$$
\begin{align}
\Chi(\varphi) &= \int_{P} s(x) \cdot \cos\left( 2\pi \tfrac{n}{P} x -\varphi \right)\, dx\quad \varphi \in \left[ 0, 2\pi \right]\\
&=\cos(\varphi)
\underbrace{\int_{P} s(x) \cdot \cos\left( 2\pi \tfrac{n}{P} x\right) dx}_{X(0)}
+
\sin(\varphi) \underbrace{\int_{P} s(x) \cdot \sin\left( 2\pi \tfrac{n}{P} x\right) dx}_{ X(\pi/2) }
\end{align}
$$
Fortunately, it is not necessary to evaluate this entire function, because its derivative is zero at the maximum:
$$
X'(\varphi) = \sin(\varphi)\cdot X(0) - \cos(\varphi)\cdot X(\pi/2) = 0, \quad \textrm{at}\ \varphi = \varphi_n.
$$
Hence
$$
\varphi_n \equiv \arctan(b_n/a_n) = \arctan(X(\pi/2)/X(0)).
$$
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### Common notations
The notation
$$
c_n
$$
is inadequate for discussing the Fourier coefficients of several different functions. Therefore, it is customarily replaced by a modified form of the function (
$$
s,
$$
in this case), such as
$$
\widehat{s}(n)
$$
or
$$
S[n],
$$
and functional notation often replaces subscripting:
$$
\begin{align}
s(x) &= \sum_{n=-\infty}^\infty \widehat{s}(n)\cdot e^{i 2\pi \tfrac{n}{P} x} && \scriptstyle \text{common mathematics notation} \\
&= \sum_{n=-\infty}^\infty S[n]\cdot e^{i 2\pi \tfrac{n}{P} x} && \scriptstyle \text{common engineering notation}
\end{align}
$$
In engineering, particularly when the variable
$$
x
$$
represents time, the coefficient sequence is called a frequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.
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Therefore, it is customarily replaced by a modified form of the function (
$$
s,
$$
in this case), such as
$$
\widehat{s}(n)
$$
or
$$
S[n],
$$
and functional notation often replaces subscripting:
$$
\begin{align}
s(x) &= \sum_{n=-\infty}^\infty \widehat{s}(n)\cdot e^{i 2\pi \tfrac{n}{P} x} && \scriptstyle \text{common mathematics notation} \\
&= \sum_{n=-\infty}^\infty S[n]\cdot e^{i 2\pi \tfrac{n}{P} x} && \scriptstyle \text{common engineering notation}
\end{align}
$$
In engineering, particularly when the variable
$$
x
$$
represents time, the coefficient sequence is called a frequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.
Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:
$$
S(f) \ \triangleq \ \sum_{n=-\infty}^\infty S[n]\cdot \delta \left(f-\frac{n}{P}\right),
$$
where
$$
f
$$
represents a continuous frequency domain.
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Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.
Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:
$$
S(f) \ \triangleq \ \sum_{n=-\infty}^\infty S[n]\cdot \delta \left(f-\frac{n}{P}\right),
$$
where
$$
f
$$
represents a continuous frequency domain. When variable
$$
x
$$
has units of seconds, _ BLOCK9_ has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of
$$
\tfrac{1}{P}
$$
, which is called the fundamental frequency. _BLOCK11
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The "teeth" of the comb are spaced at multiples (i.e. harmonics) of
$$
\tfrac{1}{P}
$$
, which is called the fundamental frequency. _BLOCK11 _ can be recovered from this representation by an inverse Fourier transform:
$$
\begin{align}
\mathcal{F}^{-1}\{S(f)\} &= \int_{-\infty}^\infty \left( \sum_{n=-\infty}^\infty S[n]\cdot \delta \left(f-\frac{n}{P}\right)\right) e^{i 2 \pi f x}\,df, \\[6pt]
&= \sum_{n=-\infty}^\infty S[n]\cdot \int_{-\infty}^\infty \delta\left(f-\frac{n}{P}\right) e^{i 2 \pi f x}\,df, \\[6pt]
&= \sum_{n=-\infty}^\infty S[n]\cdot e^{i 2\pi \tfrac{n}{P} x} \ \ \triangleq \ s(x).
\end{align}
$$
The constructed function
$$
S(f)
$$
is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.
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_BLOCK11 _ can be recovered from this representation by an inverse Fourier transform:
$$
\begin{align}
\mathcal{F}^{-1}\{S(f)\} &= \int_{-\infty}^\infty \left( \sum_{n=-\infty}^\infty S[n]\cdot \delta \left(f-\frac{n}{P}\right)\right) e^{i 2 \pi f x}\,df, \\[6pt]
&= \sum_{n=-\infty}^\infty S[n]\cdot \int_{-\infty}^\infty \delta\left(f-\frac{n}{P}\right) e^{i 2 \pi f x}\,df, \\[6pt]
&= \sum_{n=-\infty}^\infty S[n]\cdot e^{i 2\pi \tfrac{n}{P} x} \ \ \triangleq \ s(x).
\end{align}
$$
The constructed function
$$
S(f)
$$
is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.
## Table of common Fourier series
Some common pairs of periodic functions and their Fourier series coefficients are shown in the table below.
-
$$
s(x)
$$
designates a periodic function with period
$$
P.
$$
-
$$
a_0, a_n, b_n
$$
designate the Fourier series coefficients (sine-cosine form) of the periodic function
$$
s(x).
$$
Time domain PlotFrequency domain (sine-cosine form) RemarksReferenceFull-wave rectified sineHalf-wave rectified sine
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_ can be recovered from this representation by an inverse Fourier transform:
$$
\begin{align}
\mathcal{F}^{-1}\{S(f)\} &= \int_{-\infty}^\infty \left( \sum_{n=-\infty}^\infty S[n]\cdot \delta \left(f-\frac{n}{P}\right)\right) e^{i 2 \pi f x}\,df, \\[6pt]
&= \sum_{n=-\infty}^\infty S[n]\cdot \int_{-\infty}^\infty \delta\left(f-\frac{n}{P}\right) e^{i 2 \pi f x}\,df, \\[6pt]
&= \sum_{n=-\infty}^\infty S[n]\cdot e^{i 2\pi \tfrac{n}{P} x} \ \ \triangleq \ s(x).
\end{align}
$$
The constructed function
$$
S(f)
$$
is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.
## Table of common Fourier series
Some common pairs of periodic functions and their Fourier series coefficients are shown in the table below.
-
$$
s(x)
$$
designates a periodic function with period
$$
P.
$$
-
$$
a_0, a_n, b_n
$$
designate the Fourier series coefficients (sine-cosine form) of the periodic function
$$
s(x).
$$
Time domain PlotFrequency domain (sine-cosine form) RemarksReferenceFull-wave rectified sineHalf-wave rectified sine
## Table of basic transformation rules
This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients.
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## Table of common Fourier series
Some common pairs of periodic functions and their Fourier series coefficients are shown in the table below.
-
$$
s(x)
$$
designates a periodic function with period
$$
P.
$$
-
$$
a_0, a_n, b_n
$$
designate the Fourier series coefficients (sine-cosine form) of the periodic function
$$
s(x).
$$
Time domain PlotFrequency domain (sine-cosine form) RemarksReferenceFull-wave rectified sineHalf-wave rectified sine
## Table of basic transformation rules
This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients. Notation:
- Complex conjugation is denoted by an asterisk.
-
$$
s(x),r(x)
$$
designate
$$
P
$$
-periodic functions or functions defined only for
$$
x \in [0,P].
$$
-
$$
S[n], R[n]
$$
designate the Fourier series coefficients (exponential form) of
$$
s
$$
and
$$
r.
$$
Property Time domain Frequency domain (exponential form) Remarks Reference Linearity Time reversal / Frequency reversal Time conjugation Time reversal & conjugation Real part in time Imaginary part in time Real part in frequency Imaginary part in frequency Shift in time /
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Notation:
- Complex conjugation is denoted by an asterisk.
-
$$
s(x),r(x)
$$
designate
$$
P
$$
-periodic functions or functions defined only for
$$
x \in [0,P].
$$
-
$$
S[n], R[n]
$$
designate the Fourier series coefficients (exponential form) of
$$
s
$$
and
$$
r.
$$
Property Time domain Frequency domain (exponential form) Remarks Reference Linearity Time reversal / Frequency reversal Time conjugation Time reversal & conjugation Real part in time Imaginary part in time Real part in frequency Imaginary part in frequency Shift in time / Modulation in frequency Shift in frequency / Modulation in time
## Properties
### Symmetry relations
When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO.
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## Properties
### Symmetry relations
When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:
$$
\begin{array}{rlcccccccc}
\mathsf{Time\ domain} & s & = & s_{\mathrm{RE}} & + & s_{\mathrm{RO}} & + & i\ s_{\mathrm{IE}} & + & i\ s_{\mathrm{IO}} \\
&\Bigg\Updownarrow\mathcal{F} & &\Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F}\\
\mathsf{Frequency\ domain} & S & = & S_\mathrm{RE} & + & i\ S_\mathrm{IO}\, & + & i\ S_\mathrm{IE} & + & S_\mathrm{RO}
\end{array}
$$
From this, various relationships are apparent, for example:
- The transform of a real-valued function _
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### Symmetry relations
When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:
$$
\begin{array}{rlcccccccc}
\mathsf{Time\ domain} & s & = & s_{\mathrm{RE}} & + & s_{\mathrm{RO}} & + & i\ s_{\mathrm{IE}} & + & i\ s_{\mathrm{IO}} \\
&\Bigg\Updownarrow\mathcal{F} & &\Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F}\\
\mathsf{Frequency\ domain} & S & = & S_\mathrm{RE} & + & i\ S_\mathrm{IO}\, & + & i\ S_\mathrm{IE} & + & S_\mathrm{RO}
\end{array}
$$
From this, various relationships are apparent, for example:
- The transform of a real-valued function _ BLOCK1_ is the conjugate symmetric function
$$
S_\mathrm{RE}+i\ S_\mathrm{IO}.
$$
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And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:
$$
\begin{array}{rlcccccccc}
\mathsf{Time\ domain} & s & = & s_{\mathrm{RE}} & + & s_{\mathrm{RO}} & + & i\ s_{\mathrm{IE}} & + & i\ s_{\mathrm{IO}} \\
&\Bigg\Updownarrow\mathcal{F} & &\Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F}\\
\mathsf{Frequency\ domain} & S & = & S_\mathrm{RE} & + & i\ S_\mathrm{IO}\, & + & i\ S_\mathrm{IE} & + & S_\mathrm{RO}
\end{array}
$$
From this, various relationships are apparent, for example:
- The transform of a real-valued function _ BLOCK1_ is the conjugate symmetric function
$$
S_\mathrm{RE}+i\ S_\mathrm{IO}.
$$
Conversely, a conjugate symmetric transform implies a real-valued time-domain.
- The transform of an imaginary-valued function
$$
(i\ s_\mathrm{IE}+i\ s_\mathrm{IO})
$$
is the conjugate antisymmetric function
$$
S_\mathrm{RO}+i\ S_\mathrm{IE},
$$
and the converse is true.
- The transform of a conjugate symmetric function
$$
(s_\mathrm{RE}+i\ s_\mathrm{IO})
$$
is the real-valued function
$$
S_\mathrm{RE}+S_\mathrm{RO},
$$
and the converse is true.
- The transform of a conjugate antisymmetric function _
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BLOCK1_ is the conjugate symmetric function
$$
S_\mathrm{RE}+i\ S_\mathrm{IO}.
$$
Conversely, a conjugate symmetric transform implies a real-valued time-domain.
- The transform of an imaginary-valued function
$$
(i\ s_\mathrm{IE}+i\ s_\mathrm{IO})
$$
is the conjugate antisymmetric function
$$
S_\mathrm{RO}+i\ S_\mathrm{IE},
$$
and the converse is true.
- The transform of a conjugate symmetric function
$$
(s_\mathrm{RE}+i\ s_\mathrm{IO})
$$
is the real-valued function
$$
S_\mathrm{RE}+S_\mathrm{RO},
$$
and the converse is true.
- The transform of a conjugate antisymmetric function _ BLOCK7_ is the imaginary-valued function
$$
i\ S_\mathrm{IE}+i\ S_\mathrm{IO},
$$
and the converse is true.
### Riemann–Lebesgue lemma
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BLOCK7_ is the imaginary-valued function
$$
i\ S_\mathrm{IE}+i\ S_\mathrm{IO},
$$
and the converse is true.
### Riemann–Lebesgue lemma
If
$$
S
$$
is integrable,
$$
\lim_{|n| \to \infty} S[n]=0
$$
,
$$
\lim_{n \to +\infty} a_n=0
$$
and _ BLOCK3_Parseval's theorem
If
$$
s
$$
belongs to
$$
L^2(P)
$$
(periodic over an interval of length
$$
P
$$
) then:
$$
\frac{1}{P}\int_{P} |s(x)|^2 \, dx = \sum_{n=-\infty}^\infty \Bigl|S[n]\Bigr|^2.
$$
### Plancherel's theorem
If
$$
c_0,\, c_{\pm 1},\, c_{\pm 2}, \ldots
$$
are coefficients and
$$
\sum_{n=-\infty}^\infty |c_n|^2 < \infty
$$
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then:
$$
\frac{1}{P}\int_{P} |s(x)|^2 \, dx = \sum_{n=-\infty}^\infty \Bigl|S[n]\Bigr|^2.
$$
### Plancherel's theorem
If
$$
c_0,\, c_{\pm 1},\, c_{\pm 2}, \ldots
$$
are coefficients and
$$
\sum_{n=-\infty}^\infty |c_n|^2 < \infty
$$
then there is a unique function
$$
s\in L^2(P)
$$
such that
$$
S[n] = c_n
$$
for every
$$
n
$$
.
### Convolution theorems
Given
$$
P
$$
-periodic functions,
$$
s_P
$$
and
$$
r_P
$$
with Fourier series coefficients
$$
S[n]
$$
and
$$
R[n],
$$
$$
n \in \mathbb{Z},
$$
-
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then there is a unique function
$$
s\in L^2(P)
$$
such that
$$
S[n] = c_n
$$
for every
$$
n
$$
.
### Convolution theorems
Given
$$
P
$$
-periodic functions,
$$
s_P
$$
and
$$
r_P
$$
with Fourier series coefficients
$$
S[n]
$$
and
$$
R[n],
$$
$$
n \in \mathbb{Z},
$$
- The pointwise product:
$$
h_P(x) \triangleq s_P(x)\cdot r_P(x)
$$
is also
$$
P
$$
-periodic, and its Fourier series coefficients are given by the discrete convolution of the
$$
S
$$
and
$$
R
$$
sequences:
$$
H[n] = \{S*R\}[n].
$$
- The periodic convolution:
$$
h_P(x) \triangleq \int_{P} s_P(\tau)\cdot r_P(x-\tau)\, d\tau
$$
is also
$$
P
$$
-periodic, with Fourier series coefficients: _
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The pointwise product:
$$
h_P(x) \triangleq s_P(x)\cdot r_P(x)
$$
is also
$$
P
$$
-periodic, and its Fourier series coefficients are given by the discrete convolution of the
$$
S
$$
and
$$
R
$$
sequences:
$$
H[n] = \{S*R\}[n].
$$
- The periodic convolution:
$$
h_P(x) \triangleq \int_{P} s_P(\tau)\cdot r_P(x-\tau)\, d\tau
$$
is also
$$
P
$$
-periodic, with Fourier series coefficients: _ BLOCK13_- A doubly infinite sequence
$$
\left \{c_n \right \}_{n \in Z}
$$
in
$$
c_0(\mathbb{Z})
$$
is the sequence of Fourier coefficients of a function in
$$
L^1([0,2\pi])
$$
if and only if it is a convolution of two sequences in
$$
\ell^2(\mathbb{Z})
$$
. See
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if and only if it is a convolution of two sequences in
$$
\ell^2(\mathbb{Z})
$$
. See
### Derivative property
If
$$
s
$$
is a 2-periodic function on
$$
\mathbb{R}
$$
which is
$$
k
$$
times differentiable, and its
$$
k^{\text{th}}
$$
derivative is continuous, then
$$
s
$$
belongs to the function space
$$
C^k(\mathbb{R})
$$
.
-
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### Derivative property
If
$$
s
$$
is a 2-periodic function on
$$
\mathbb{R}
$$
which is
$$
k
$$
times differentiable, and its
$$
k^{\text{th}}
$$
derivative is continuous, then
$$
s
$$
belongs to the function space
$$
C^k(\mathbb{R})
$$
.
- If
$$
s \in C^k(\mathbb{R})
$$
, then the Fourier coefficients of the
$$
k^{\text{th}}
$$
derivative of
$$
s
$$
can be expressed in terms of the Fourier coefficients
$$
\widehat{s}[n]
$$
of
$$
s
$$
, via the formula
$$
\widehat{s^{(k)}}[n] = (in)^k \widehat{s}[n].
$$
In particular, since for any fixed
$$
k\geq 1
$$
we have
$$
\widehat{s^{(k)}}[n]\to 0
$$
as
$$
n\to\infty
$$
, it follows that
$$
|n|^k\widehat{s}[n]
$$
tends to zero, i.e., the Fourier coefficients converge to zero faster than the
$$
k^{\text{th}}
$$
power of
$$
|n|
$$
.
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- If
$$
s \in C^k(\mathbb{R})
$$
, then the Fourier coefficients of the
$$
k^{\text{th}}
$$
derivative of
$$
s
$$
can be expressed in terms of the Fourier coefficients
$$
\widehat{s}[n]
$$
of
$$
s
$$
, via the formula
$$
\widehat{s^{(k)}}[n] = (in)^k \widehat{s}[n].
$$
In particular, since for any fixed
$$
k\geq 1
$$
we have
$$
\widehat{s^{(k)}}[n]\to 0
$$
as
$$
n\to\infty
$$
, it follows that
$$
|n|^k\widehat{s}[n]
$$
tends to zero, i.e., the Fourier coefficients converge to zero faster than the
$$
k^{\text{th}}
$$
power of
$$
|n|
$$
.
### Compact groups
One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. If that is the property which we seek to preserve, one can produce Fourier series on any compact group. Typical examples include those classical groups that are compact.
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If that is the property which we seek to preserve, one can produce Fourier series on any compact group. Typical examples include those classical groups that are compact. This generalizes the Fourier transform to all spaces of the form L2(G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products. The Fourier series exists and converges in similar ways to the case.
An alternative extension to compact groups is the Peter–Weyl theorem, which proves results about representations of compact groups analogous to those about finite groups.
### Riemannian manifolds
If the domain is not a group, then there is no intrinsically defined convolution. However, if
$$
X
$$
is a compact Riemannian manifold, it has a Laplace–Beltrami operator. The Laplace–Beltrami operator is the differential operator that corresponds to Laplace operator for the Riemannian manifold
$$
X
$$
. Then, by analogy, one can consider heat equations on
$$
X
$$
. Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace–Beltrami operator as a basis. This generalizes Fourier series to spaces of the type
$$
L^2(X)
$$
, where
$$
X
$$
is a Riemannian manifold.
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Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace–Beltrami operator as a basis. This generalizes Fourier series to spaces of the type
$$
L^2(X)
$$
, where
$$
X
$$
is a Riemannian manifold. The Fourier series converges in ways similar to the
$$
[-\pi,\pi]
$$
case. A typical example is to take
$$
X
$$
to be the sphere with the usual metric, in which case the Fourier basis consists of spherical harmonics.
### Locally compact Abelian groups
The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. However, there is a straightforward generalization to Locally Compact Abelian (LCA) groups.
This generalizes the Fourier transform to
$$
L^1(G)
$$
or
$$
L^2(G)
$$
, where
$$
G
$$
is an LCA group. If
$$
G
$$
is compact, one also obtains a Fourier series, which converges similarly to the
$$
[-\pi,\pi]
$$
case, but if
$$
G
$$
is noncompact, one obtains instead a Fourier integral.
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This generalizes the Fourier transform to
$$
L^1(G)
$$
or
$$
L^2(G)
$$
, where
$$
G
$$
is an LCA group. If
$$
G
$$
is compact, one also obtains a Fourier series, which converges similarly to the
$$
[-\pi,\pi]
$$
case, but if
$$
G
$$
is noncompact, one obtains instead a Fourier integral. This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is
$$
\mathbb{R}
$$
.
## Extensions
### Fourier-Stieltjes series
Let
$$
F(x)
$$
be a function of bounded variation defined on the closed interval
$$
[0,P]\subseteq\mathbb{R}
$$
. The Fourier series whose coefficients are given by
$$
c_n = \frac{1}{P}\int_0^P \ e^{-i 2\pi \tfrac{n}{P} x }\,dF(x), \quad \forall n\in\mathbb{Z},
$$
is called the Fourier-Stieltjes series. The space of functions of bounded variation
$$
BV
$$
is a subspace of
$$
L^1
$$
.
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The Fourier series whose coefficients are given by
$$
c_n = \frac{1}{P}\int_0^P \ e^{-i 2\pi \tfrac{n}{P} x }\,dF(x), \quad \forall n\in\mathbb{Z},
$$
is called the Fourier-Stieltjes series. The space of functions of bounded variation
$$
BV
$$
is a subspace of
$$
L^1
$$
. As any
$$
F \in BV
$$
defines a Radon measure (i.e. a locally finite Borel measure on
$$
\mathbb{R}
$$
), this definition can be extended as follows.
Consider the space
$$
M
$$
of all finite Borel measures on the real line; as such
$$
L^1 \subset M
$$
. If there is a measure
$$
\mu \in M
$$
such that the Fourier-Stieltjes coefficients are given by
$$
c_n = \hat\mu(n)=\frac{1}{P}\int_0^P \ e^{-i 2\pi \tfrac{n}{P} x }\,d\mu(x), \quad \forall n\in\mathbb{Z},
$$
then the series is called a Fourier-Stieltjes series.
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Consider the space
$$
M
$$
of all finite Borel measures on the real line; as such
$$
L^1 \subset M
$$
. If there is a measure
$$
\mu \in M
$$
such that the Fourier-Stieltjes coefficients are given by
$$
c_n = \hat\mu(n)=\frac{1}{P}\int_0^P \ e^{-i 2\pi \tfrac{n}{P} x }\,d\mu(x), \quad \forall n\in\mathbb{Z},
$$
then the series is called a Fourier-Stieltjes series. Likewise, the function
$$
\hat\mu(n)
$$
, where
$$
\mu \in M
$$
, is called a Fourier-Stieltjes transform.
The question whether or not
$$
\mu
$$
exists for a given sequence of
$$
c_n
$$
forms the basis of the trigonometric moment problem.
Furthermore,
$$
M
$$
is a strict subspace of the space of (tempered) distributions
$$
\mathcal{D}
$$
, i.e.,
$$
M \subset \mathcal{D}
$$
.
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The question whether or not
$$
\mu
$$
exists for a given sequence of
$$
c_n
$$
forms the basis of the trigonometric moment problem.
Furthermore,
$$
M
$$
is a strict subspace of the space of (tempered) distributions
$$
\mathcal{D}
$$
, i.e.,
$$
M \subset \mathcal{D}
$$
. If the Fourier coefficients are determined by a distribution
$$
F \in \mathcal{D}
$$
then the series is described as a Fourier-Schwartz series. Contrary to the Fourier-Stieltjes series, deciding whether a given series is a Fourier series or a Fourier-Schwartz series is relatively trivial due to the characteristics of its dual space; the Schwartz space
$$
\mathcal{S}(\mathbb{R}^n)
$$
.
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If the Fourier coefficients are determined by a distribution
$$
F \in \mathcal{D}
$$
then the series is described as a Fourier-Schwartz series. Contrary to the Fourier-Stieltjes series, deciding whether a given series is a Fourier series or a Fourier-Schwartz series is relatively trivial due to the characteristics of its dual space; the Schwartz space
$$
\mathcal{S}(\mathbb{R}^n)
$$
.
### Fourier series on a square
We can also define the Fourier series for functions of two variables
$$
x
$$
and
$$
y
$$
in the square
$$
[-\pi,\pi]\times[-\pi,\pi]
$$
:
$$
\begin{align}
f(x,y) & = \sum_{j,k \in \Z} c_{j,k}e^{ijx}e^{iky},\\[5pt]
c_{j,k} & = \frac{1}{4 \pi^2} \int_{-\pi}^\pi \int_{-\pi}^\pi f(x,y) e^{-ijx}e^{-iky}\, dx \, dy.
\end{align}
$$
Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression.
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Contrary to the Fourier-Stieltjes series, deciding whether a given series is a Fourier series or a Fourier-Schwartz series is relatively trivial due to the characteristics of its dual space; the Schwartz space
$$
\mathcal{S}(\mathbb{R}^n)
$$
.
### Fourier series on a square
We can also define the Fourier series for functions of two variables
$$
x
$$
and
$$
y
$$
in the square
$$
[-\pi,\pi]\times[-\pi,\pi]
$$
:
$$
\begin{align}
f(x,y) & = \sum_{j,k \in \Z} c_{j,k}e^{ijx}e^{iky},\\[5pt]
c_{j,k} & = \frac{1}{4 \pi^2} \int_{-\pi}^\pi \int_{-\pi}^\pi f(x,y) e^{-ijx}e^{-iky}\, dx \, dy.
\end{align}
$$
Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression. In particular, the JPEG image compression standard uses the two-dimensional discrete cosine transform, a discrete form of the Fourier cosine transform, which uses only cosine as the basis function.
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### Fourier series on a square
We can also define the Fourier series for functions of two variables
$$
x
$$
and
$$
y
$$
in the square
$$
[-\pi,\pi]\times[-\pi,\pi]
$$
:
$$
\begin{align}
f(x,y) & = \sum_{j,k \in \Z} c_{j,k}e^{ijx}e^{iky},\\[5pt]
c_{j,k} & = \frac{1}{4 \pi^2} \int_{-\pi}^\pi \int_{-\pi}^\pi f(x,y) e^{-ijx}e^{-iky}\, dx \, dy.
\end{align}
$$
Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression. In particular, the JPEG image compression standard uses the two-dimensional discrete cosine transform, a discrete form of the Fourier cosine transform, which uses only cosine as the basis function.
For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.
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In particular, the JPEG image compression standard uses the two-dimensional discrete cosine transform, a discrete form of the Fourier cosine transform, which uses only cosine as the basis function.
For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.
### Fourier series of a Bravais-lattice-periodic function
A three-dimensional Bravais lattice is defined as the set of vectors of the form
$$
\mathbf{R} = n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3
$$
where _ BLOCK1_ are integers and
$$
\mathbf{a}_i
$$
are three linearly independent but not necessarily orthogonal vectors. Let us consider some function
$$
f(\mathbf{r})
$$
with the same periodicity as the Bravais lattice, i.e.
$$
f(\mathbf{r}) = f(\mathbf{R}+\mathbf{r})
$$
for any lattice vector
$$
\mathbf{R}
$$
. This situation frequently occurs in solid-state physics where
$$
f(\mathbf{r})
$$
might, for example, represent the effective potential that an electron "feels" inside a periodic crystal.
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Let us consider some function
$$
f(\mathbf{r})
$$
with the same periodicity as the Bravais lattice, i.e.
$$
f(\mathbf{r}) = f(\mathbf{R}+\mathbf{r})
$$
for any lattice vector
$$
\mathbf{R}
$$
. This situation frequently occurs in solid-state physics where
$$
f(\mathbf{r})
$$
might, for example, represent the effective potential that an electron "feels" inside a periodic crystal. In presence of such a periodic potential, the quantum-mechanical description of the electron results in a periodically modulated plane-wave commonly known as Bloch state.
In order to develop
$$
f(\mathbf{r})
$$
in a Fourier series, it is convenient to introduce an auxiliary function
$$
g(x_1,x_2,x_3) \triangleq f(\mathbf{r}) = f \left (x_1\frac{\mathbf{a}_{1}}{a_1}+x_2\frac{\mathbf{a}_{2}}{a_2}+x_3\frac{\mathbf{a}_{3}}{a_3} \right ).
$$
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In presence of such a periodic potential, the quantum-mechanical description of the electron results in a periodically modulated plane-wave commonly known as Bloch state.
In order to develop
$$
f(\mathbf{r})
$$
in a Fourier series, it is convenient to introduce an auxiliary function
$$
g(x_1,x_2,x_3) \triangleq f(\mathbf{r}) = f \left (x_1\frac{\mathbf{a}_{1}}{a_1}+x_2\frac{\mathbf{a}_{2}}{a_2}+x_3\frac{\mathbf{a}_{3}}{a_3} \right ).
$$
Both
$$
f(\mathbf{r})
$$
and
$$
g(x_1,x_2,x_3)
$$
contain essentially the same information.
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In order to develop
$$
f(\mathbf{r})
$$
in a Fourier series, it is convenient to introduce an auxiliary function
$$
g(x_1,x_2,x_3) \triangleq f(\mathbf{r}) = f \left (x_1\frac{\mathbf{a}_{1}}{a_1}+x_2\frac{\mathbf{a}_{2}}{a_2}+x_3\frac{\mathbf{a}_{3}}{a_3} \right ).
$$
Both
$$
f(\mathbf{r})
$$
and
$$
g(x_1,x_2,x_3)
$$
contain essentially the same information. However, instead of the position vector
$$
\mathbf{r}
$$
, the arguments of
$$
g
$$
are coordinates
$$
x_{1,2,3}
$$
along the unit vectors
$$
\mathbf{a}_{i}/{a_i}
$$
of the Bravais lattice, such that
$$
g
$$
is an ordinary periodic function in these variables,
$$
g(x_1,x_2,x_3) = g(x_1+a_1,x_2,x_3) = g(x_1,x_2+a_2,x_3) = g(x_1,x_2,x_3+a_3)\quad\forall\;x_1,x_2,x_3.
$$
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Both
$$
f(\mathbf{r})
$$
and
$$
g(x_1,x_2,x_3)
$$
contain essentially the same information. However, instead of the position vector
$$
\mathbf{r}
$$
, the arguments of
$$
g
$$
are coordinates
$$
x_{1,2,3}
$$
along the unit vectors
$$
\mathbf{a}_{i}/{a_i}
$$
of the Bravais lattice, such that
$$
g
$$
is an ordinary periodic function in these variables,
$$
g(x_1,x_2,x_3) = g(x_1+a_1,x_2,x_3) = g(x_1,x_2+a_2,x_3) = g(x_1,x_2,x_3+a_3)\quad\forall\;x_1,x_2,x_3.
$$
This trick allows us to develop
$$
g
$$
as a multi-dimensional Fourier series, in complete analogy with the square-periodic function discussed in the previous section.
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However, instead of the position vector
$$
\mathbf{r}
$$
, the arguments of
$$
g
$$
are coordinates
$$
x_{1,2,3}
$$
along the unit vectors
$$
\mathbf{a}_{i}/{a_i}
$$
of the Bravais lattice, such that
$$
g
$$
is an ordinary periodic function in these variables,
$$
g(x_1,x_2,x_3) = g(x_1+a_1,x_2,x_3) = g(x_1,x_2+a_2,x_3) = g(x_1,x_2,x_3+a_3)\quad\forall\;x_1,x_2,x_3.
$$
This trick allows us to develop
$$
g
$$
as a multi-dimensional Fourier series, in complete analogy with the square-periodic function discussed in the previous section. Its Fourier coefficients are
$$
\begin{align}
c(m_1, m_2, m_3)
= \frac{1}{a_3}\int_0^{a_3} dx_3 \frac{1}{a_2}\int_0^{a_2} dx_2 \frac{1}{a_1}\int_0^{a_1} dx_1\, g(x_1, x_2, x_3)\, e^{-i 2\pi \left(\tfrac{m_1}{a_1} x_1+\tfrac{m_2}{a_2} x_2 + \tfrac{m_3}{a_3} x_3\right)}
\end{align},
$$
where
$$
m_1,m_2,m_3
$$
are all integers.
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This trick allows us to develop
$$
g
$$
as a multi-dimensional Fourier series, in complete analogy with the square-periodic function discussed in the previous section. Its Fourier coefficients are
$$
\begin{align}
c(m_1, m_2, m_3)
= \frac{1}{a_3}\int_0^{a_3} dx_3 \frac{1}{a_2}\int_0^{a_2} dx_2 \frac{1}{a_1}\int_0^{a_1} dx_1\, g(x_1, x_2, x_3)\, e^{-i 2\pi \left(\tfrac{m_1}{a_1} x_1+\tfrac{m_2}{a_2} x_2 + \tfrac{m_3}{a_3} x_3\right)}
\end{align},
$$
where
$$
m_1,m_2,m_3
$$
are all integers.
$$
c(m_1,m_2,m_3)
$$
plays the same role as the coefficients
$$
c_{j,k}
$$
in the previous section but in order to avoid double subscripts we note them as a function.
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Its Fourier coefficients are
$$
\begin{align}
c(m_1, m_2, m_3)
= \frac{1}{a_3}\int_0^{a_3} dx_3 \frac{1}{a_2}\int_0^{a_2} dx_2 \frac{1}{a_1}\int_0^{a_1} dx_1\, g(x_1, x_2, x_3)\, e^{-i 2\pi \left(\tfrac{m_1}{a_1} x_1+\tfrac{m_2}{a_2} x_2 + \tfrac{m_3}{a_3} x_3\right)}
\end{align},
$$
where
$$
m_1,m_2,m_3
$$
are all integers.
$$
c(m_1,m_2,m_3)
$$
plays the same role as the coefficients
$$
c_{j,k}
$$
in the previous section but in order to avoid double subscripts we note them as a function.
Once we have these coefficients, the function
$$
g
$$
can be recovered via the Fourier series
$$
g(x_1, x_2, x_3)=\sum_{m_1, m_2, m_3 \in \Z } \,c(m_1, m_2, m_3) \, e^{i 2\pi \left( \tfrac{m_1}{a_1} x_1+ \tfrac{m_2}{a_2} x_2 + \tfrac{m_3}{a_3} x_3\right)}.
$$
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$$
c(m_1,m_2,m_3)
$$
plays the same role as the coefficients
$$
c_{j,k}
$$
in the previous section but in order to avoid double subscripts we note them as a function.
Once we have these coefficients, the function
$$
g
$$
can be recovered via the Fourier series
$$
g(x_1, x_2, x_3)=\sum_{m_1, m_2, m_3 \in \Z } \,c(m_1, m_2, m_3) \, e^{i 2\pi \left( \tfrac{m_1}{a_1} x_1+ \tfrac{m_2}{a_2} x_2 + \tfrac{m_3}{a_3} x_3\right)}.
$$
We would now like to abandon the auxiliary coordinates
$$
x_{1,2,3}
$$
and to return to the original position vector
$$
\mathbf{r}
$$
.
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Once we have these coefficients, the function
$$
g
$$
can be recovered via the Fourier series
$$
g(x_1, x_2, x_3)=\sum_{m_1, m_2, m_3 \in \Z } \,c(m_1, m_2, m_3) \, e^{i 2\pi \left( \tfrac{m_1}{a_1} x_1+ \tfrac{m_2}{a_2} x_2 + \tfrac{m_3}{a_3} x_3\right)}.
$$
We would now like to abandon the auxiliary coordinates
$$
x_{1,2,3}
$$
and to return to the original position vector
$$
\mathbf{r}
$$
. This can be achieved by means of the reciprocal lattice whose vectors
$$
\mathbf{b}_{1,2,3}
$$
are defined such that they are orthonormal (up to a factor
$$
2\pi
$$
) to the original Bravais vectors
$$
\mathbf{a}_{1,2,3}
$$
,
$$
\mathbf{a}_i\cdot\mathbf{b_j}=2\pi\delta_{ij},
$$
with
$$
\delta_{ij}
$$
the Kronecker delta.
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We would now like to abandon the auxiliary coordinates
$$
x_{1,2,3}
$$
and to return to the original position vector
$$
\mathbf{r}
$$
. This can be achieved by means of the reciprocal lattice whose vectors
$$
\mathbf{b}_{1,2,3}
$$
are defined such that they are orthonormal (up to a factor
$$
2\pi
$$
) to the original Bravais vectors
$$
\mathbf{a}_{1,2,3}
$$
,
$$
\mathbf{a}_i\cdot\mathbf{b_j}=2\pi\delta_{ij},
$$
with
$$
\delta_{ij}
$$
the Kronecker delta. With this, the scalar product between a reciprocal lattice vector
$$
\mathbf{Q}
$$
and an arbitrary position vector
$$
\mathbf{r}
$$
written in the Bravais lattice basis becomes
$$
\mathbf{Q} \cdot \mathbf{r} = \left ( m_1\mathbf{b}_1 + m_2\mathbf{b}_2 + m_3\mathbf{b}_3 \right ) \cdot \left (x_1\frac{\mathbf{a}_1}{a_1}+ x_2\frac{\mathbf{a}_2}{a_2} +x_3\frac{\mathbf{a}_3}{a_3} \right ) = 2\pi \left( x_1\frac{m_1}{a_1}+x_2\frac{m_2}{a_2}+x_3\frac{m_3}{a_3} \right ),
$$
which is exactly the expression occurring in the Fourier exponents.
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|
The Fourier series for
$$
f(\mathbf{r}) =g(x_1,x_2,x_3)
$$
can therefore be rewritten as a sum over the all reciprocal lattice vectors
$$
\mathbf{Q}= m_1\mathbf{b}_1+m_2\mathbf{b}_2+m_3\mathbf{b}_3
$$
,
$$
f(\mathbf{r})=\sum_{\mathbf{Q}} c(\mathbf{Q})\, e^{i \mathbf{Q} \cdot \mathbf{r}},
$$
and the coefficients are
$$
c(\mathbf{Q}) = \frac{1}{a_3} \int_0^{a_3} dx_3 \, \frac{1}{a_2}\int_0^{a_2} dx_2 \, \frac{1}{a_1}\int_0^{a_1} dx_1 \, f\left(x_1\frac{\mathbf{a}_1}{a_1} + x_2\frac{\mathbf{a}_2}{a_2} + x_3\frac{\mathbf{a}_3}{a_3} \right) e^{-i \mathbf{Q} \cdot \mathbf{r}}.
$$
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|
The remaining task will be to convert this integral over lattice coordinates back into a volume integral.
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https://en.wikipedia.org/wiki/Fourier_series
|
The relation between the lattice coordinates
$$
x_{1,2,3}
$$
and the original cartesian coordinates
$$
\mathbf{r} = (x,y,z)
$$
is a linear system of equations,
$$
\mathbf{r} = x_1\frac{\mathbf{a}_1}{a_1}+x_2\frac{\mathbf{a}_2}{a_2}+x_3\frac{\mathbf{a}_3}{a_3},
$$
which, when written in matrix form,
$$
\begin{bmatrix}x\\y\\z\end{bmatrix} =\mathbf{J}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} =\begin{bmatrix}\frac{\mathbf{a}_1}{a_1},\frac{\mathbf{a}_2}{a_2},\frac{\mathbf{a}_3}{a_3}\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\,,
$$
involves a constant matrix
$$
\mathbf{J}
$$
whose columns are the unit vectors
$$
\mathbf{a}_j/a_j
$$
of the Bravais lattice.
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|
When changing variables from
$$
\mathbf{r}
$$
to
$$
(x_1,x_2,x_3)
$$
in an integral, the same matrix
$$
\mathbf{J}
$$
appears as a Jacobian matrix
$$
\mathbf{J}=\begin{bmatrix}
\dfrac{\partial x}{\partial x_1} & \dfrac{\partial x}{\partial x_2} & \dfrac{\partial x}{\partial x_3
} \\[12pt]
\dfrac{\partial y}{\partial x_1} & \dfrac{\partial y}{\partial x_2} & \dfrac{\partial y}{\partial x_3} \\[12pt]
\dfrac{\partial z}{\partial x_1} & \dfrac{\partial z}{\partial x_2} & \dfrac{\partial z}{\partial x_3}
\end{bmatrix}\,.
$$
Its determinant
$$
J
$$
is therefore also constant and can be inferred from any integral over any domain; here we choose to calculate the volume of the primitive unit cell
$$
\Gamma
$$
in both coordinate systems:
$$
V_{\Gamma} = \int_{\Gamma} d^3 r = J \int_{0}^{a_1} dx_1 \int_{0}^{a_2} dx_2 \int_{0}^{a_3} dx_3=J\, a_1 a_2 a_3
$$
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|
The unit cell being a parallelepiped, we have
$$
V_{\Gamma}=\mathbf{a}_1\cdot(\mathbf{a}_2 \times \mathbf{a}_3)
$$
and thus
$$
d^3r=J dx_1 dx_2 dx_3 =\frac{\mathbf{a}_1\cdot(\mathbf{a}_2 \times \mathbf{a}_3)}{a_1 a_2 a_3} dx_1 dx_2 dx_3.
$$
This allows us to write
$$
c (\mathbf{Q})
$$
as the desired volume integral over the primitive unit cell
$$
\Gamma
$$
in ordinary cartesian coordinates:
$$
c(\mathbf{Q}) = \frac{1}{\mathbf{a}_1\cdot(\mathbf{a}_2 \times \mathbf{a}_3)}\int_{\Gamma} d^3 r\, f(\mathbf{r})\cdot e^{-i \mathbf{Q} \cdot \mathbf{r}}\,.
$$
### Hilbert space
As the trigonometric series is a special class of orthogonal system, Fourier series can naturally be defined in the context of Hilbert spaces.
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|
This allows us to write
$$
c (\mathbf{Q})
$$
as the desired volume integral over the primitive unit cell
$$
\Gamma
$$
in ordinary cartesian coordinates:
$$
c(\mathbf{Q}) = \frac{1}{\mathbf{a}_1\cdot(\mathbf{a}_2 \times \mathbf{a}_3)}\int_{\Gamma} d^3 r\, f(\mathbf{r})\cdot e^{-i \mathbf{Q} \cdot \mathbf{r}}\,.
$$
### Hilbert space
As the trigonometric series is a special class of orthogonal system, Fourier series can naturally be defined in the context of Hilbert spaces. For example, the space of square-integrable functions on
$$
[-\pi,\pi]
$$
forms the Hilbert space
$$
L^2([-\pi,\pi])
$$
.
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|
### Hilbert space
As the trigonometric series is a special class of orthogonal system, Fourier series can naturally be defined in the context of Hilbert spaces. For example, the space of square-integrable functions on
$$
[-\pi,\pi]
$$
forms the Hilbert space
$$
L^2([-\pi,\pi])
$$
. Its inner product, defined for any two elements
$$
f
$$
and
$$
g
$$
, is given by:
$$
\langle f, g \rangle = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\overline{g(x)}\,dx.
$$
This space is equipped with the orthonormal basis
$$
\left\{e_n=e^{inx}: n \in \Z\right\}
$$
.
Then the (generalized)
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|
Its inner product, defined for any two elements
$$
f
$$
and
$$
g
$$
, is given by:
$$
\langle f, g \rangle = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\overline{g(x)}\,dx.
$$
This space is equipped with the orthonormal basis
$$
\left\{e_n=e^{inx}: n \in \Z\right\}
$$
.
Then the (generalized) Fourier series expansion of
$$
f \in L^{2}([-\pi,\pi])
$$
, given by
$$
f(x) = \sum_{n=-\infty}^\infty c_n e^{i n x },
$$
can be written as
$$
f=\sum_{n=-\infty}^\infty \langle f,e_n \rangle \, e_n.
$$
The sine-cosine form follows in a similar fashion.
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|
Indeed, the sines and cosines form an orthogonal set:
$$
\int_{-\pi}^{\pi} \cos(mx)\, \cos(nx)\, dx = \frac{1}{2}\int_{-\pi}^{\pi} \cos((n-m)x)+\cos((n+m)x)\, dx = \pi \delta_{mn}, \quad m, n \ge 1,
$$
$$
\int_{-\pi}^{\pi} \sin(mx)\, \sin(nx)\, dx = \frac{1}{2}\int_{-\pi}^{\pi} \cos((n-m)x)-\cos((n+m)x)\, dx = \pi \delta_{mn}, \quad m, n \ge 1
$$
(where δmn is the Kronecker delta), and
$$
\int_{-\pi}^{\pi} \cos(mx)\, \sin(nx)\, dx = \frac{1}{2}\int_{-\pi}^{\pi} \sin((n+m)x)+\sin((n-m)x)\, dx = 0;
$$
Hence, the set
$$
\left\{\frac{1}{\sqrt{2}},\frac{\cos x}{\sqrt{2}},\frac{\sin x}{\sqrt{2}},\dots,\frac{\cos (nx)}{\sqrt{2}},\frac{\sin (nx)}{\sqrt{2}},\dots \right\},
$$
also forms an orthonormal basis for
$$
L^2([-\pi,\pi])
$$
.
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|
The density of their span is a consequence of the Stone–Weierstrass theorem, but follows also from the properties of classical kernels like the Fejér kernel.
## Fourier theorem proving convergence of Fourier series
In engineering, the Fourier series is generally assumed to converge except at jump discontinuities since the functions encountered in engineering are usually better-behaved than those in other disciplines. In particular, if
$$
s
$$
is continuous and the derivative of
$$
s(x)
$$
(which may not exist everywhere) is square integrable, then the Fourier series of
$$
s
$$
converges absolutely and uniformly to
$$
s(x)
$$
. If a function is square-integrable on the interval
$$
[x_0,x_0+P]
$$
, then the Fourier series converges to the function almost everywhere. It is possible to define Fourier coefficients for more general functions or distributions, in which case pointwise convergence often fails, and convergence in norm or weak convergence is usually studied.
The theorems proving that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions), and informal variations of them that do not specify the convergence conditions, are sometimes referred to generically as Fourier's theorem or the Fourier theorem.
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|
It is possible to define Fourier coefficients for more general functions or distributions, in which case pointwise convergence often fails, and convergence in norm or weak convergence is usually studied.
The theorems proving that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions), and informal variations of them that do not specify the convergence conditions, are sometimes referred to generically as Fourier's theorem or the Fourier theorem.
### Least squares property
The earlier :
$$
s_N(x) = \sum_{n=-N}^N S[n]\ e^{i 2\pi\tfrac{n}{P} x},
$$
is a trigonometric polynomial of degree _ BLOCK1_ that can be generally expressed as:
$$
p_N(x)=\sum_{n=-N}^N p[n]\ e^{i 2\pi\tfrac{n}{P}x}.
$$
Parseval's theorem implies that:
### Convergence theorems
Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.
If
$$
s
$$
is continuously differentiable, then _ BLOCK1_ is the
$$
n^{\text{th}}
$$
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|
If
$$
s
$$
is continuously differentiable, then _ BLOCK1_ is the
$$
n^{\text{th}}
$$
Fourier coefficient of the first derivative
$$
s'
$$
. Since
$$
s'
$$
is continuous, and therefore bounded, it is square-integrable and its Fourier coefficients are square-summable. Then, by the Cauchy–Schwarz inequality,
$$
\left(\sum_{n\ne 0}|S[n]|\right)^2\le \sum_{n\ne 0}\frac1{n^2}\cdot\sum_{n\ne 0} |nS[n]|^2.
$$
This means that _ BLOCK6_ is absolutely summable. The sum of this series is a continuous function, equal to
$$
s
$$
, since the Fourier series converges in
$$
L^1
$$
to
$$
s
$$
:
This result can be proven easily if
$$
s
$$
is further assumed to be
$$
C^2
$$
, since in that case
$$
n^2S[n]
$$
tends to zero as
$$
n \rightarrow \infty
$$
.
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|
The sum of this series is a continuous function, equal to
$$
s
$$
, since the Fourier series converges in
$$
L^1
$$
to
$$
s
$$
:
This result can be proven easily if
$$
s
$$
is further assumed to be
$$
C^2
$$
, since in that case
$$
n^2S[n]
$$
tends to zero as
$$
n \rightarrow \infty
$$
. More generally, the Fourier series is absolutely summable, thus converges uniformly to
$$
s
$$
, provided that
$$
s
$$
satisfies a Hölder condition of order
$$
\alpha > 1/2
$$
. In the absolutely summable case, the inequality:
$$
\sup_x |s(x) - s_N(x)| \le \sum_{|n| > N} |S[n]|
$$
proves uniform convergence.
Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at
$$
x
$$
if
$$
s
$$
is differentiable at
$$
x
$$
, to more sophisticated results such as Carleson's theorem which states that the Fourier series of an
$$
L^2
$$
function converges almost everywhere.
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|
In the absolutely summable case, the inequality:
$$
\sup_x |s(x) - s_N(x)| \le \sum_{|n| > N} |S[n]|
$$
proves uniform convergence.
Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at
$$
x
$$
if
$$
s
$$
is differentiable at
$$
x
$$
, to more sophisticated results such as Carleson's theorem which states that the Fourier series of an
$$
L^2
$$
function converges almost everywhere.
### Divergence
Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous T-periodic function need not converge pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact.
In 1922, Andrey Kolmogorov published an article titled Une série de Fourier-Lebesgue divergente presque partout in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere.
|
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|
In 1922, Andrey Kolmogorov published an article titled Une série de Fourier-Lebesgue divergente presque partout in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere.
It is possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in [0,π] by
$$
f(x) = \sum_{n=1}^{\infty} \frac{1}{n^2} \sin\left[ \left( 2^{n^3} +1 \right) \frac{x}{2}\right].
$$
Because the function is even the Fourier series contains only cosines:
$$
\sum_{m=0}^\infty C_m \cos(mx).
$$
The coefficients are:
$$
C_m=\frac 1\pi\sum_{n=1}^{\infty} \frac{1}{n^2} \left\{\frac 2{2^{n^3} +1-2m}+\frac 2{2^{n^3} +1+2m}\right\}
$$
As increases, the coefficients will be positive and increasing until they reach a value of about
$$
C_m\approx 2/(n^2\pi)
$$
at
$$
m=2^{n^3}/2
$$
for some and then become negative (starting with a value around
$$
-2/(n^2\pi)
$$
) and getting smaller, before starting a new such wave.
|
https://en.wikipedia.org/wiki/Fourier_series
|
He later constructed an example of an integrable function whose Fourier series diverges everywhere.
It is possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in [0,π] by
$$
f(x) = \sum_{n=1}^{\infty} \frac{1}{n^2} \sin\left[ \left( 2^{n^3} +1 \right) \frac{x}{2}\right].
$$
Because the function is even the Fourier series contains only cosines:
$$
\sum_{m=0}^\infty C_m \cos(mx).
$$
The coefficients are:
$$
C_m=\frac 1\pi\sum_{n=1}^{\infty} \frac{1}{n^2} \left\{\frac 2{2^{n^3} +1-2m}+\frac 2{2^{n^3} +1+2m}\right\}
$$
As increases, the coefficients will be positive and increasing until they reach a value of about
$$
C_m\approx 2/(n^2\pi)
$$
at
$$
m=2^{n^3}/2
$$
for some and then become negative (starting with a value around
$$
-2/(n^2\pi)
$$
) and getting smaller, before starting a new such wave. At
$$
x=0
$$
the Fourier series is simply the running sum of
$$
C_m,
$$
and this builds up to around
$$
\frac 1{n^2\pi}\sum_{k=0}^{2^{n^3}/2}\frac 2{2k+1}\sim\frac 1{n^2\pi}\ln 2^{n^3}=\frac n\pi\ln 2
$$
in the th wave before returning to around zero, showing that the series does not converge at zero but reaches higher and higher peaks.
|
https://en.wikipedia.org/wiki/Fourier_series
|
It is possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in [0,π] by
$$
f(x) = \sum_{n=1}^{\infty} \frac{1}{n^2} \sin\left[ \left( 2^{n^3} +1 \right) \frac{x}{2}\right].
$$
Because the function is even the Fourier series contains only cosines:
$$
\sum_{m=0}^\infty C_m \cos(mx).
$$
The coefficients are:
$$
C_m=\frac 1\pi\sum_{n=1}^{\infty} \frac{1}{n^2} \left\{\frac 2{2^{n^3} +1-2m}+\frac 2{2^{n^3} +1+2m}\right\}
$$
As increases, the coefficients will be positive and increasing until they reach a value of about
$$
C_m\approx 2/(n^2\pi)
$$
at
$$
m=2^{n^3}/2
$$
for some and then become negative (starting with a value around
$$
-2/(n^2\pi)
$$
) and getting smaller, before starting a new such wave. At
$$
x=0
$$
the Fourier series is simply the running sum of
$$
C_m,
$$
and this builds up to around
$$
\frac 1{n^2\pi}\sum_{k=0}^{2^{n^3}/2}\frac 2{2k+1}\sim\frac 1{n^2\pi}\ln 2^{n^3}=\frac n\pi\ln 2
$$
in the th wave before returning to around zero, showing that the series does not converge at zero but reaches higher and higher peaks. Note that though the function is continuous, it is not differentiable.
|
https://en.wikipedia.org/wiki/Fourier_series
|
Linear hashing (LH) is a dynamic data structure which implements a hash table and grows or shrinks one bucket at a time. It was invented by Witold Litwin in 1980.
It has been analyzed by Baeza-Yates and Soza-Pollman. It is the first in a number of schemes known as dynamic hashing
such as Larson's Linear Hashing with Partial Extensions, Linear Hashing with Priority Splitting, Linear Hashing with Partial Expansions and Priority Splitting, or Recursive Linear Hashing.
The file structure of a dynamic hashing data structure adapts itself to changes in the size of the file, so expensive periodic file reorganization is avoided. A Linear Hashing file expands by splitting a predetermined bucket into two and shrinks by merging two predetermined buckets into one. The trigger for a reconstruction depends on the flavor of the scheme; it could be an overflow at a bucket or load factor (i.e., the number of records divided by the number of buckets) moving outside of a predetermined range. In Linear Hashing there are two types of buckets, those that are to be split and those already split.
|
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|
The trigger for a reconstruction depends on the flavor of the scheme; it could be an overflow at a bucket or load factor (i.e., the number of records divided by the number of buckets) moving outside of a predetermined range. In Linear Hashing there are two types of buckets, those that are to be split and those already split. While extendible hashing splits only overflowing buckets, spiral hashing (a.k.a. spiral storage) distributes records unevenly over the buckets such that buckets with high costs of insertion, deletion, or retrieval are earliest in line for a split.
Linear Hashing has also been made into a scalable distributed data structure,
### LH*
. In LH*, each bucket resides at a different server. LH* itself has been expanded to provide data availability in the presence of failed buckets. Key based operations (inserts, deletes, updates, reads) in LH and LH* take maximum constant time independent of the number of buckets and hence of records.
## Algorithm details
Records in LH or LH* consists of a key and a content, the latter basically all the other attributes of the record. They are stored in buckets. For example, in Ellis' implementation, a bucket is a linked list of records.
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|
They are stored in buckets. For example, in Ellis' implementation, a bucket is a linked list of records. The file allows the key based CRUD operations create or insert, read, update, and delete as well as a scan operations that scans all records, for example to do a database select operation on a non-key attribute. Records are stored in buckets whose numbering starts with 0.
The key distinction from schemes such as Fagin's extendible hashing is that as the file expands due to insertions, only one bucket is split at a time, and the order in which buckets are split is already predetermined.
### Hash functions
The hash function
$$
h_i(c)
$$
returns the 0-based index of the bucket that contains the record with key
$$
c
$$
. When a bucket which uses the hash function _ BLOCK3_ is split into two new buckets, the hash function
$$
h_i
$$
is replaced with
$$
h_{i+1}
$$
for both of those new buckets. At any time, at most two hash functions
$$
h_l
$$
and
$$
h_{l+1}
$$
are used; such that
$$
l
$$
corresponds to the current level. The family of hash functions
$$
h_i(c)
$$
is also referred to as the dynamic hash function.
|
https://en.wikipedia.org/wiki/Linear_hashing
|
At any time, at most two hash functions
$$
h_l
$$
and
$$
h_{l+1}
$$
are used; such that
$$
l
$$
corresponds to the current level. The family of hash functions
$$
h_i(c)
$$
is also referred to as the dynamic hash function.
Typically, the value of
$$
i
$$
in
$$
h_i
$$
corresponds to the number of rightmost binary digits of the key
$$
c
$$
that are used to segregate the buckets. This dynamic hash function can be expressed arithmetically as
$$
h_i(c) \mapsto (c \bmod 2^i)
$$
. Note that when the total number of buckets is equal to one,
$$
i=0
$$
.
Complete the calculations below to determine the correct hashing function for the given hashing key
$$
c
$$
.
```python
1. l represents the current level
1. s represents the split pointer index
a = h_l(c)
if (a < s): a = h_{l+1}(c)
```
### Split control
Linear hashing algorithms may use only controlled splits or both controlled and uncontrolled splits.
Controlled splitting occurs if a split is performed whenever the load factor, which is monitored by the file, exceeds a predetermined threshold.
|
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|
### Split control
Linear hashing algorithms may use only controlled splits or both controlled and uncontrolled splits.
Controlled splitting occurs if a split is performed whenever the load factor, which is monitored by the file, exceeds a predetermined threshold. If the hash index uses controlled splitting, the buckets are allowed to overflow by using linked overflow blocks. When the load factor surpasses a set threshold, the split pointer's designated bucket is split. Instead of using the load factor, this threshold can also be expressed as an occupancy percentage, in which case, the maximum number of records in the hash index equals (occupancy percentage)*(max records per non-overflowed bucket)*(number of buckets).
An uncontrolled split occurs when a split is performed whenever a bucket overflows, in which case that bucket would be split into two separate buckets.
File contraction occurs in some LH algorithm implementations if a controlled split causes the load factor to sink below a threshold. In this case, a merge operation would be triggered which would undo the last split, and reset the file state.
### Split pointer
The index of the next bucket to be split is part of the file state and called the split pointer
$$
s
$$
.
|
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|
In this case, a merge operation would be triggered which would undo the last split, and reset the file state.
### Split pointer
The index of the next bucket to be split is part of the file state and called the split pointer
$$
s
$$
. The split pointer corresponds to the first bucket that uses the hash function
$$
h_l
$$
instead of
$$
h_{l+1}
$$
.
For example, if numerical records are inserted into the hash index according to their farthest right binary digits, the bucket corresponding with the appended bucket will be split. Thus, if we have the buckets labelled as 000, 001, 10, 11, 100, 101, we would split the bucket 10 because we are appending and creating the next sequential bucket 110. This would give us the buckets 000, 001, 010, 11, 100, 101, 110.
|
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|
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