Poisson summation formula

In mathematics, the Poisson summation formula is an equation that allows us to relate 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

For appropriate functions ƒ, the Poisson summation formula may be stated as:

$\sum _{n=-\infty }^{\infty }f(n)=\sum _{k=-\infty }^{\infty }{\hat {f}}\left(k\right),$ (Eq.1)

where ${\hat {f}}$  is the Fourier transform of $f\,$;  i.e. ${\hat {f}}(\nu )={\mathcal {F}}\{f(x)\}.$

With the substitution, $g(nT)\ {\stackrel {\text{def}}{=}}\ f(n),\,$  and the Fourier transform property,  ${\mathcal {F}}\{g(xT)\}\ ={\frac {1}{T}}\cdot {\hat {g}}\left({\frac {\nu }{T}}\right)$  (for T > 0),  Eq.1 becomes (Stein & Weiss 1971):

$\sum _{n=-\infty }^{\infty }g(nT)={\frac {1}{T}}\sum _{k=-\infty }^{\infty }{\hat {g}}\left({\frac {k}{T}}\right).$ (Eq.2)

The periodic summation (with period T) result is obtained by defining,  $s(t+nT)\ {\stackrel {\text{def}}{=}}\ g(nT)\,$   and applying the transform property,  ${\mathcal {F}}\{s(t+x)\}\ ={\hat {s}}(\nu )\cdot \exp \left(i2\pi \nu t\right).$  Then Eq.2 becomes (Pinsky 2002; Zygmund 1968):

$\underbrace {\sum _{n=-\infty }^{\infty }s(t+nT)} _{S_{T}(t)}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }{\hat {s}}\left({\frac {k}{T}}\right)\ \exp \left(i2\pi {\frac {k}{T}}t\right).$ (Eq.3)

Applicability

A simple demonstration of the feasibility of Eq.1 is as follows:

${\begin{aligned}\sum _{n=-\infty }^{\infty }f(n)&=\sum _{n=-\infty }^{\infty }\left(\int _{-\infty }^{\infty }{\hat {f}}(x)\ e^{i2\pi nx}dx\right)\\&=\int _{-\infty }^{\infty }{\hat {f}}(x)\left(\sum _{n=-\infty }^{\infty }e^{i2\pi nx}\right)dx\end{aligned}}$

The quantity in parentheses is zero for all values of x except those of the form x=k, where k is any integer. At those values the summation diverges at a rate that is independent of k. The summation can effectively be replaced by an infinite sequence of equal-strength Dirac delta functions (called Dirac comb), and we continue accordingly.

${\begin{aligned}\sum _{n=-\infty }^{\infty }f(n)&=\int _{-\infty }^{\infty }{\hat {f}}(x)\left(\sum _{k=-\infty }^{\infty }\delta (x-k)\right)dx\\&=\sum _{k=-\infty }^{\infty }\underbrace {\left(\int _{-\infty }^{\infty }{\hat {f}}(x)\ \delta (x-k)\ dx\right)} _{{\hat {f}}(k)}\end{aligned}}$

Conditions that ensure Eq.3 is applicable are that ƒ is a continuous integrable function which satisfies

$|f(x)|+|{\hat {f}}(x)|\leq C(1+|x|)^{-1-\delta }$

for some C, δ > 0 and every x (Grafakos 2004; Stein & Weiss 1971). Note that such ƒ is uniformly continuous; this together with the decay assumption on ƒ, show that the series defining S_T converges uniformly to a continuous function. Eq.3 holds in the strong sense that both sides converge uniformly and absolutely to the same limit (Stein & Weiss 1971).

Eq.3 also holds in a pointwise sense under the strictly weaker assumption that ƒ has bounded variation and

$2f(x)=\lim _{\varepsilon \to 0}f(x+\varepsilon )+\lim _{\varepsilon \to 0}f(x-\varepsilon ).$   (Zygmund 1968)

The Fourier series on the right-hand side of Eq.3 is then understood as a (conditionally convergent) limit of symmetric partial sums.

Eq.3 holds under the much less restrictive assumption that ƒ is in L¹(R), but then it is necessary to interpret it in the sense that the right-hand side is the (possibly divergent) Fourier series of S_T(t) (Zygmund 1968). 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 holds under the less restrictive conditions that ƒ is integrable and 0 is a point of continuity of S_T(t). However Eq.2 may fail to hold even when both $f\,$ and ${\hat {f}}$ are integrable and continuous, and the sums converge absolutely (Katznelson 1976).

Eq.3 holds in the sense that if ƒ ∈ L¹(R), then the right-hand side is the (possibly divergent) Fourier series of the left-hand side. This proof may be found in either (Pinsky 2002) or (Zygmund 1968). It follows from the dominated convergence theorem that S_T(t) exists and is finite for almost every t and furthermore it follows that S_T is integrable on the interval [0,T].

Applications

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 R² 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 (Grafakos 2004).

In signal processing, the Poisson summation formula leads to the Nyquist–Shannon sampling theorem (Pinsky 2002).

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. (A broad function in real space becomes a narrow function in Fourier space and vice versa.) This is the essential idea behind Ewald summation.

The Poisson summation formula may be used to derive Landau's asymptotic formula for the number of lattice points in a large Euclidean sphere. It can also be used to show that if an integrable function, $f\,$ and ${\hat {f}}$ both have compact support then $f=0\,$  (Pinsky 2002).

Poisson summation can also be used to derive a variety of functional equations including the functional equation for the Riemann zeta function.

Distributional formulation

Eq.2 can be interpreted in the language of distributions (Córdoba 1988; Hörmander 1983, §7.2). Let δ(t) be the Dirac delta function. Define

$\Delta _{T}(t)\ =\ \sum _{n=-\infty }^{\infty }\delta (t-nT)$

summed over all integers n. It can be shown that Δ_T is a tempered distribution, known as the Dirac comb: intuitively, this follows since applied to any Schwartz function one gets a bi-infinite series whose tails decay rapidly. One may then interpret the summation formula as the formula:

$\Delta _{T}={\frac {1}{T}}{\hat {\Delta }}_{T}.$ (Eq.4)

That is, up to an overall normalization, Δ_T is its own Fourier transform. Indeed, if ƒ is a Schwartz function, then applying Eq.4 to ƒ gives precisely Eq.2; alternatively, taking a convolution with ƒ gives precisely Eq.3. If ƒ is a function or distribution that is sufficiently regular that such a convolution can be defined in the distribution sense, then Eq.3 holds in the sense of distributions.

Generalizations

A version of the Poisson summation formula holds in Euclidean space with only slight modifications. Let Λ be the lattice in R^d consisting of points with integer coordinates. Then for suitable functions ƒ, one has

$\sum _{\nu \in \Lambda }f(x+\nu )=\sum _{\nu \in \Lambda }{\hat {f}}(\nu )e^{2\pi ix\cdot \nu }.$    (Stein & Weiss 1971, VII §2)

As in the case of one variable, this holds in the sense of Fourier series if ƒ is assumed to be integrable, and pointwise if ƒ satisfies the decay condition

$|f(x)|+|{\hat {f}}(x)|\leq C(1+|x|)^{-d-\delta }$

for some C, δ > 0.

More generally, a version of the statement holds if Λ is replaced by a more general lattice in R^d. The dual lattice Λ′ can be defined as a subset of the dual vector space or alternatively by Pontryagin duality. Then the statement is that the sum of delta-functions at each point of Λ, and at each point of Λ′, are again Fourier transforms as distributions, subject to correct normalization.

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.

Further generalisation to locally compact abelian groups is required in number theory. In non-commutative harmonic analysis, the idea is taken even further in the Selberg trace formula, but takes on a much deeper character.

See also

• Relations between Fourier transforms and Fourier series

References

• Benedetto, J.J.; Zimmermann, G. (1997), "Sampling multipliers and the Poisson summation formula", J. Fourier Ana. App., 3 (5).
• Córdoba, A., "La formule sommatoire de Poisson", C.R. Acad. Sci. Paris, Series I, 306: 373–376.
• Higgins, J.R. (1985), "Five short stories about the cardinal series", Bull. AMS, 12 (1): 45–89, doi:10.1090/S0273-0979-1985-15293-0.
• Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., vol. 256, Springer, ISBN 3-540-12104-8, MR0717035.
• Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Pearson Education, Inc., pp. 253–257, ISBN 0-13-035399-X.
• Katznelson, Yitzhak (1976), An introduction to harmonic analysis (Second corrected ed.), New York: Dover Publications, Inc, ISBN 0-486-63331-4
• Pinsky, M. (2002), Introduction to Fourier Analysis and Wavelets., Brooks Cole.
• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9.
• Zygmund, Antoni (1968), Trigonometric series (2nd ed.), Cambridge University Press (published 1988), ISBN 978-0521358859.