Poisson Summation: Where Sampling Meets Periodization

1. The precise statement

Let f: ℝ → ℂ and define its Fourier transform f̂(ξ) = ∫_ℝ f(x) e^−2πixξ dx. The Poisson summation formula (PSF) asserts

∑_n∈ℤ f(n) = ∑_k∈ℤ f̂(k).

More generally, evaluating at a shift x gives the periodized identity ∑_n∈ℤ f(x+n) = ∑_k∈ℤ f̂(k) e^2πikx, of which the first line is the case x = 0.

Hypotheses matter. A clean sufficient condition (due in this sharp form to work around Zygmund): f is continuous, and there exist C > 0, δ > 0 with |f(x)| ≤ C(1+|x|)^−1−δ and |f̂(ξ)| ≤ C(1+|ξ|)^−1−δ. Then both series converge absolutely and the identity holds pointwise, everywhere. If f is Schwartz (smooth, rapidly decaying with all derivatives), every hypothesis is automatic. Absolute convergence of both sides is not a mere convenience — it is what makes the periodization continuous and its Fourier series converge to it.

2. The picture: periodization is sampling in disguise

Take f and wrap it around a circle of circumference 1 by adding all its integer translates: F(x) = ∑_n f(x+n). This periodization F is 1-periodic by construction. As a function on the circle ℝ/ℤ it has Fourier coefficients — and a one-line computation shows those coefficients are exactly the samples f̂(k) of the original transform.

So the picture is a duality: folding the space axis into a period-1 loop corresponds, on the Fourier side, to reading f̂ only at integer frequencies. Compression in one domain (period 1) forces discretization in the other (spacing 1). Halve the period and the frequency samples spread out by two; this is literally the Nyquist relation. The two grids ℤ (space translations) and ℤ (frequency samples) are dual: each is the annihilator of the other under e^2πi(·), which is why the identity is exact rather than approximate.

3. Key idea of the proof

The mechanism is: periodize, then expand in a Fourier series, then evaluate at 0.

Define F(x) = ∑_n∈ℤ f(x+n). Under the decay hypothesis this series converges absolutely and uniformly on compact sets, so F is continuous and 1-periodic. Its Fourier coefficients are

c_k = ∫₀¹ F(x) e^−2πikx dx = ∫₀¹ ∑_n f(x+n) e^−2πikx dx.

Interchange sum and integral (justified by absolute convergence) and unfold the shifted intervals [n, n+1) back into all of ℝ — the unfolding trick — to get c_k = ∫_ℝ f(x) e^−2πikx dx = f̂(k). Hence F(x) = ∑_k f̂(k) e^2πikx, valid pointwise because the coefficients f̂(k) are absolutely summable (that is where the decay of f̂ is used). Setting x = 0 gives ∑_n f(n) = ∑_k f̂(k). The clever step is entirely the unfolding: summing over n and integrating over one period reassembles the whole line.

4. Canonical example: the Gaussian and Jacobi's theta identity

Let f_t(x) = e^−πtx² for t > 0, the fixed shape of the Fourier transform. Its transform is f̂_t(ξ) = t^−1/2 e^−πξ²/t. Plugging into PSF:

∑_n∈ℤ e^−πtn² = t^−1/2 ∑_k∈ℤ e^−πk²/t.

Writing θ(t) = ∑_n e^−πtn², this is precisely Jacobi's transformation law θ(t) = t^−1/2 θ(1/t) — the modularity of the theta function under t ↦ 1/t. Numerically at t = 1 both sides equal θ(1) ≈ 1.0864 and the factor t^−1/2 = 1 makes the symmetry a self-duality. This one identity, fed through the Mellin transform ∫₀^∞ (θ(t)−1)/2 · t^s/2−1 dt, yields the analytic continuation and functional equation of the Riemann zeta function ζ(s) = ζ(1−s) (completed). Poisson summation is the engine underneath.

5. Why the hypotheses matter — and what breaks

Decay alone is not enough; pointwise care is real. A famous cautionary example: take a continuous f with ∑_n f(n) and ∑_k f̂(k) both convergent yet unequal, which is possible when the periodization's Fourier series converges but not to the value one expects at a discontinuity, or when convergence is only conditional. If f is merely integrable but its transform decays too slowly, ∑_k f̂(k) may diverge or the interchange of sum and integral fails — the unfolding step is exactly where an unjustified swap poisons the identity.

A sharp positive result (Katznelson): if f is continuous, of bounded variation, and integrable with f̂ integrable, PSF holds. The moral is that both f and f̂ must be controlled; controlling only one side is a classic trap. Connections run deep: PSF is Fourier-series duality (fourier-series) applied to the circle ℝ/ℤ, and on a locally compact abelian group it is the statement that a lattice and its Pontryagin dual pair perfectly.

6. Applications and significance

Sampling theory. The Nyquist–Shannon theorem is Poisson summation read backwards: sampling a signal at rate 1/T periodizes its spectrum with period 1/T, so a band-limited signal is recoverable iff those spectral copies don't overlap. Aliasing is literally the overlap terms in ∑_k f̂(ξ − k/T).

Number theory. Beyond ζ's functional equation, PSF over a lattice Λ ⊂ ℝⁿ (with dual Λ* and covolume) gives Epstein zeta functional equations, controls lattice point counts, and underlies the recent sphere-packing solutions in dimensions 8 and 24 (Viazovska 2016), where a magic function built to satisfy Poisson summation constraints proves optimality.

Physics and computation. In crystallography the diffraction pattern of a lattice is its dual lattice — pure Poisson summation. Ewald summation for electrostatics, and fast evaluation of heat kernels on tori, all exploit that a slowly converging space-side sum equals a rapidly converging frequency-side sum, or vice versa. Choosing which side to compute is a superpower.

How period/spacing, lattice, and dual object transform under Poisson summation

Setting	Space side	Frequency side	Identity
Integers ℤ	period 1 periodization	samples at integers	∑ₙ f(n) = ∑ₖ f̂(k)
Scaled ℤ, spacing T	period-T periodization	samples spaced 1/T	T∑ₙ f(nT) = ∑ₖ f̂(k/T)
Lattice Λ ⊂ ℝⁿ	sum over Λ	sum over dual Λ*	∑_{v∈Λ} f(v) = (1/covol Λ) ∑_{w∈Λ*} f̂(w)
Shifted / general x	periodization at x	modulated samples	∑ₙ f(x+n) = ∑ₖ f̂(k) e^{2πikx}
Gaussian, width t	∑ e^{−πtn²}	∑ e^{−πk²/t}/√t	θ(1/t) = √t · θ(t)