Poisson summation formula
Generated by GPT-5-miniExpansion Funnel Raw 49 → Dedup 0 → NER 0 → Enqueued 0
| Poisson summation formula | |
|---|---|
| Name | Poisson summation formula |
| Field | Mathematics |
| Introduced | 19th century |
| Contributors | Siméon Denis Poisson, Joseph Fourier, Peter Gustav Lejeune Dirichlet |
Poisson summation formula
The Poisson summation formula relates sums of a function over integer lattices to sums of its Fourier transform, connecting discrete and continuous analysis and appearing across harmonic analysis, number theory, and signal processing. It plays a central role in work by Siméon Denis Poisson, Joseph Fourier, Bernhard Riemann, Srinivasa Ramanujan, and John von Neumann, and links classical topics like the Fourier transform, the theta function, and the Riemann zeta function.
Statement
In its classical form for a sufficiently nice function f: R → C, the formula states that the sum over integers of f equals the sum over integers of the Fourier transform F of f, up to normalization conventions used by Joseph Fourier and later authors such as Hermann Weyl and André Weil. Variants are written with different normalizations employed in works by Hardy, G. H. Hardy, John Littlewood, and D. H. Lehmer. A common normalization expresses the relation as an equality between Σ_{n∈Z} f(n) and Σ_{k∈Z} F(k), where F denotes the Fourier transform defined in the style of Jean-Baptiste Joseph Fourier and adopted by Paul Dirac in distribution theory.
Proofs and derivations
Proofs originate in classical analysis and the theory of distributions developed by Laurent Schwartz. Early derivations trace to expansions of the Jacobi theta function used by Carl Gustav Jacobi and were systematized by Siméon Denis Poisson and Peter Gustav Lejeune Dirichlet. Modern proofs include: - A proof via Fourier series on the torus, using orthogonality relations found in texts by Bernhard Riemann and Georg Cantor. - A distributional proof invoking the machinery of Laurent Schwartz with influences from work by Paul Dirac and André Weil. - Proofs using complex analysis and contour integration techniques developed by Augustin-Louis Cauchy and Bernhard Riemann in his study of the zeta function.
Expositions by E. T. Whittaker, G. N. Watson, and Elias Stein connect these approaches, while general functional-analytic derivations refer to nuclear spaces and theorems from Alexander Grothendieck and John von Neumann.
Applications
The formula is applied across a wide array of problems treated by Srinivasa Ramanujan and modern researchers: it underlies proofs of the functional equation for the Riemann zeta function appearing in the work of Bernhard Riemann and Atle Selberg, facilitates lattice-point counting problems studied by Hermann Minkowski and George Pólya, and informs signal reconstruction principles used in engineering by Claude Shannon and Norbert Wiener. In crystallography and solid-state physics the formula connects reciprocal lattices discussed by Max von Laue and Paul Peter Ewald. In analytic number theory it yields the Voronoi summation formula built upon ideas of G. H. Hardy and Freeman Dyson. The formula also appears in the study of modular forms by Srinivasa Ramanujan and G. H. Hardy, and in modern trace formulae initiated by Atle Selberg and extended by Robert Langlands.
Generalizations and extensions
Generalizations extend the classical formula to higher-dimensional lattices treated by Hermann Minkowski and John Milnor, to locally compact abelian groups as formulated in Pontryagin duality by Lev Pontryagin, and to nonabelian settings appearing in the nonabelian harmonic analysis developed by Harish-Chandra and Robert Langlands. The adelic and automorphic extensions used in modern number theory reference constructions of André Weil and the adelic framework influenced by Alexander Grothendieck and John Tate. Extensions include discrete-to-continuous correspondences in the work of Jean-Pierre Serre on algebraic number fields and lattice theta series explored by Carl Ludwig Siegel.
Examples and computations
Classical computations include deriving the modular transformation formula for the Jacobi theta function attributed to Carl Gustav Jacobi and applications to Gaussian sums considered by Adrien-Marie Legendre and Pierre-Simon Laplace. Concrete examples compute the sum of sampled Gaussians to produce rapidly convergent series used by Srinivasa Ramanujan in formulae for π, and derive the Poisson kernel in potential theory influenced by George Green and Siméon Denis Poisson. Numerical applications and fast algorithms connect to the Fast Fourier transform developed by James Cooley and John Tukey and to sampling theorems formalized by Claude Shannon.