The equality that relates the sum of a function sampled on a lattice in Euclidean space to the sum of its Fourier transform sampled on the dual lattice; commonly expressed for functions on R^n as sum_{m in Z^n} f(m) = sum_{k in Z^n} 36f(k), making a precise bridge between discrete sums and harmonic analysis.

Principle

A function and its Fourier transform encode dual descriptions of the same data on primal and frequency lattices; summing over one lattice equals summing the transform over the dual lattice when analytic and decay conditions permit interchange of summation and transform.

Demonstration

For the Gaussian f(x)=e^{-pi x^2} on R, summing f over Z yields the classical theta transformation: sum_{n in Z} e^{-pi n^2 t} = t^{-1/2} sum_{m in Z} e^{-pi m^2 / t}, which is a direct application of the Poisson identity after computing the Fourier transform of the Gaussian.

Misapplication

Applying the formula to functions that do not satisfy required smoothness or decay (for example, non-tempered functions or ones with divergent lattice sums) without justifying distributional interpretations leads to incorrect equalities or divergent series manipulations.

Consequence

When valid, the formula converts lattice sums into spectral sums, enabling analytic continuation, modular-type transformations, and evaluation of arithmetic sums (e.g., relating exponential sums, theta functions, and counting lattice points).

Reversal

Viewed dually, exchanging the roles of the primal lattice and the dual lattice emphasizes that information encoded in pointwise samples is equivalently encoded in frequency samples; the reversal highlights time-frequency duality rather than a one-way summation trick.

Boundary

Holds for Schwartz functions on R^n and extends to tempered distributions and suitable periodic or compactly supported test functions under additional hypotheses; it does not apply naïvely to arbitrary discontinuous or slowly decaying functions without regularization.

Semantic Tension

Competes with Euler–Maclaurin-type expansions that also relate sums and integrals asymptotically; Poisson gives an exact spectral identity under analytic hypotheses, whereas Euler–Maclaurin gives asymptotic corrections to integral approximations.

Synthesis

The Poisson Summation Formula is the precise identity equating lattice sums of a function with lattice sums of its Fourier transform; it formalizes time-frequency duality and underlies many analytic number theory and harmonic analysis results by converting discrete arithmetic problems into spectral ones.