TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Poisson summation

Statement

For every Schwartz function $f$ under $\hat{f} (ξ) = \int f (x) e (- x ξ) d x$ , $\sum_{n \in Z} f (n) = \sum_{m \in Z} \hat{f} (m) .$

Facts & Assumptions

Given: A Schwartz function $f$ .

Proof

technique · direct

1.1given

Define the periodization $F (x) = \sum_{n \in Z} f (x + n)$ . Schwartz decay makes this series, and every termwise derivative series, converge uniformly on compact sets, so $F$ is a smooth $1$ -periodic function.

2.1step 1.1

Its $m$ th Fourier coefficient is $\int_{0}^{1} F (x) e (- m x) d x = \sum_{n \in Z} \int_{0}^{1} f (x + n) e (- m (x + n)) d x = \hat{f} (m),$ where absolute convergence justifies interchange and the intervals $[n, n + 1]$ partition $R$ .

3.1step 2.1∎

The sequence $(\hat{f} (m))_{m \in Z}$ is rapidly decreasing, so the Fourier series of $F$ converges absolutely and uniformly to $F$ . Evaluating at $0$ gives $\sum_{n \in Z} f (n) = F (0) = \sum_{m \in Z} \hat{f} (m) .$

Used by

Twisted Poisson summation Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Nickolas Andersen, Analytic Number Theory, Theorem 16.5 (standard reference, not scraped)