In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.
Forms of the equation
Consider an aperiodic function with Fourier transform alternatively designated by and
A proof may be found in either Pinsky[2] or Zygmund.[3]Eq.2, for instance, holds in the sense that if , then the right-hand side is the (possibly divergent) Fourier series of the left-hand side. It follows from the dominated convergence theorem that exists and is finite for almost every . Furthermore it follows that is integrable on any interval of length So it is sufficient to show that the Fourier series coefficients of are Proceeding from the definition of the Fourier coefficients we have:
where the interchange of summation with integration is once again justified by dominated convergence. With a change of variables () this becomes:
The proof of Eq.3 is of course similar, except that the formula for series coefficients in frequency is:
The sign of is positive, because it has to be the opposite of the one on the right-hand side of Eq.3.
In other words, the periodization of a Dirac delta resulting in a Dirac comb, corresponds to the discretization of its spectrum which is constantly one.
Hence, this again is a Dirac comb but with reciprocal increments.
Eq.2 holds provided is a continuous integrable function which satisfies
for some and every [8][9] Note that such is uniformly continuous, this together with the decay assumption on , show that the series defining converges uniformly to a continuous function. Eq.2 holds in the strong sense that both sides converge uniformly and absolutely to the same limit.[9]
Eq.2 holds in a pointwise sense under the strictly weaker assumption that has bounded variation and[3]
The Fourier series on the right-hand side of Eq.2 is then understood as a (conditionally convergent) limit of symmetric partial sums.
As shown above, Eq.2 holds under the much less restrictive assumption that is in , but then it is necessary to interpret it in the sense that the right-hand side is the (possibly divergent) Fourier series of [3] In this case, one may extend the region where equality holds by considering summability methods such as Cesàro summability. When interpreting convergence in this way Eq.2, case holds under the less restrictive conditions that is integrable and 0 is a point of continuity of . However Eq.2 may fail to hold even when both and are integrable and continuous, and the sums converge absolutely.[10]
Applications
Method of images
In partial differential equations, the Poisson summation formula provides a rigorous justification for the fundamental solution of the heat equation with absorbing rectangular boundary by the method of images. Here the heat kernel on is known, and that of a rectangle is determined by taking the periodization. The Poisson summation formula similarly provides a connection between Fourier analysis on Euclidean spaces and on the tori of the corresponding dimensions.[8] In one dimension, the resulting solution is called a theta function.
In the statistical study of time-series, if is a function of time, then looking only at its values at equally spaced points of time is called "sampling." In applications, typically the function is band-limited, meaning that there is some cutoff frequency such that is zero for frequencies exceeding the cutoff: for For band-limited functions, choosing the sampling rate guarantees that no information is lost: since can be reconstructed from these sampled values. Then, by Fourier inversion, so can This leads to the Nyquist–Shannon sampling theorem.[2]
Ewald summation
Computationally, the Poisson summation formula is useful since a slowly converging summation in real space is guaranteed to be converted into a quickly converging equivalent summation in Fourier space.[12] (A broad function in real space becomes a narrow function in Fourier space and vice versa.) This is the essential idea behind Ewald summation.
Approximations of integrals
The Poisson summation formula is also useful to bound the errors obtained when an integral is approximated by a (Riemann) sum. Consider an approximation of as , where is the size of the bin. Then, according to Eq.2 this approximation coincides with . The error in the approximation can then be bounded as . This is particularly useful when the Fourier transform of is rapidly decaying if .
Lattice points inside a sphere
The Poisson summation formula may be used to derive Landau's asymptotic formula for the number of lattice points inside a large Euclidean sphere. It can also be used to show that if an integrable function, and both have compact support then [2]
Number theory
In number theory, Poisson summation can also be used to derive a variety of functional equations including the functional equation for the Riemann zeta function.[13]
One important such use of Poisson summation concerns theta functions: periodic summations of Gaussians . Put , for a complex number in the upper half plane, and define the theta function:
The relation between and turns out to be important for number theory, since this kind of relation is one of the defining properties of a modular form. By choosing and using the fact that one can conclude:
by putting
It follows from this that has a simple transformation property under and this can be used to prove Jacobi's formula for the number of different ways to express an integer as the sum of eight perfect squares.
Sphere packings
Cohn & Elkies[14] proved an upper bound on the density of sphere packings using the Poisson summation formula, which subsequently led to a proof of optimal sphere packings in dimension 8 and 24.
Other
Let for and for to get
It can be used to prove the functional equation for the theta function.
Poisson's summation formula appears in Ramanujan's notebooks and can be used to prove some of his formulas, in particular it can be used to prove one of the formulas in Ramanujan's first letter to Hardy.[clarification needed]
It can be used to calculate the quadratic Gauss sum.
Generalizations
The Poisson summation formula holds in Euclidean space of arbitrary dimension. Let be the lattice in consisting of points with integer coordinates. For a function in , consider the series given by summing the translates of by elements of :
Theorem For in , the above series converges pointwise almost everywhere, and defines a -periodic function on , hence a function on the torus a.e. lies in with
Moreover, for all in
(the Fourier transform
of on the torus ) equals
(the Fourier transform of on ).
When is in addition continuous, and both and decay sufficiently fast at infinity, then one can "invert" the Fourier series back to their domain and make a stronger statement. More precisely, if
for some C, δ > 0, then[9]: VII §2
where both series converge absolutely and uniformly on Λ. When d = 1 and x = 0, this gives Eq.1 above.
More generally, a version of the statement holds if Λ is replaced by a more general lattice in a finite dimensional vectorspace . Choose a translation invariant measure on . It is unique up to positive scalar. Again for a function we define the periodisation
as above.
The dual lattice is defined as a subset of the dual vector space that evaluates to integers on the lattice or alternatively, by Pontryagin duality, as the characters of that contain in the kernel.
Then the statement is that for all the Fourier transform of the periodisation as a function on and the Fourier transform of on itself are related by proper normalisation
Note that the right hand side is independent of the choice of invariant measure .
If and are continuous and tend to zero faster than then
In particular
This is applied in the theory of theta functions, and is a possible method in geometry of numbers. In fact in more recent work on counting lattice points in regions it is routinely used − summing the indicator function of a region D over lattice points is exactly the question, so that the LHS of the summation formula is what is sought and the RHS something that can be attacked by mathematical analysis.
A series of mathematicians applying harmonic analysis to number theory, most notably Martin Eichler, Atle Selberg, Robert Langlands, and James Arthur, have generalised the Poisson summation formula to the Fourier transform on non-commutative locally compact reductive algebraic groups with a discrete subgroup such that has finite volume. For example, can be the real points of and can be the integral points of . In this setting, plays the role of the real number line in the classical version of Poisson summation, and plays the role of the integers that appear in the sum. The generalised version of Poisson summation is called the Selberg Trace Formula, and has played a role in proving many cases of Artin's conjecture and in Wiles's proof of Fermat's Last Theorem. The left-hand side of Eq.1 becomes a sum over irreducible unitary representations of , and is called "the spectral side," while the right-hand side becomes a sum over conjugacy classes of , and is called "the geometric side."
The Poisson summation formula is the archetype for vast developments in harmonic analysis and number theory.
^Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John R. (1999). Discrete-time signal processing (2nd ed.). Upper Saddle River, N.J.: Prentice Hall. ISBN0-13-754920-2. samples of the Fourier transform of an aperiodic sequence x[n] can be thought of as DFS coefficients of a periodic sequence obtained through summing periodic replicas of x[n].
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