LemmaStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-23 (gpt-6-sol)

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Fourier partial sums are Dirichlet convolutions

Statement

Assume the Axiom of Countable Choice. Let $f$ be a one-period integrable function on $R$ . Then for every $N \geq 0$ and every $x \in R$ ,

$S_{N} f (x) = \int_{0}^{1} f (x - t) D_{N} (t) d t = (f * D_{N}) (x) .$

Facts & Assumptions

Given: Countable Choice, a one-period integrable function $f$ , an integer $N \geq 0$ , and a real $x$ .

[L1]

Fourier coefficients, Fourier partial sums, and torus convolution are defined exactly as in Period-one Fourier coefficients, partial sums, and convolution on the torus.

[L2]

The Dirichlet kernel is $D_{N} (t) = \sum_{∣ k ∣ \leq N} e_{k} (t)$ , where $e_{k} (t) = e^{2 π i k t}$ (Dirichlet and Fejer kernels).

Proof

technique · direct

1.1L1L2algebra

Expanding the convolution against the finite sum [L1, L2, algebra] gives $\int_{0}^{1} f (x - t) D_{N} (t) d t = \sum_{∣ k ∣ \leq N} \int_{0}^{1} f (x - t) e^{2 π i k t} d t .$

2.1L1step 1.1algebra

For each $k$ , substitute $u = x - t$ . Then $\int_{0}^{1} f (x - t) e^{2 π i k t} d t = e^{2 π i k x} \int_{x - 1}^{x} f (u) e^{- 2 π i k u} d u .$ The integrand $u \mapsto f (u) e^{- 2 π i k u}$ is one-periodic, so its integral over $[x - 1, x]$ equals its integral over $[0, 1]$ , namely $\hat{f} (k)$ . Therefore $\int_{0}^{1} f (x - t) e^{2 π i k t} d t = \hat{f} (k) e_{k} (x) .$

3.1L1step 2.1algebra∎

Summing step 2.1 over $∣ k ∣ \leq N$ yields $\int_{0}^{1} f (x - t) D_{N} (t) d t = \sum_{∣ k ∣ \leq N} \hat{f} (k) e_{k} (x) = S_{N} f (x) .$ By [L1], the integral is also $(f * D_{N}) (x)$ .

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard S. Laugesen, Harmonic Analysis Lecture Notes (standard reference, not scraped)
Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (standard reference, not scraped)