LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Fourier coefficients of a periodic weak derivative

Statement

If $f, g \in L^{2} (T)$ and $g = f^{'}$ in the periodic weak sense, then $\hat{g} (k) = 2 π i k \hat{f} (k) (k \in Z) .$

Facts & Assumptions

Given: $f, g \in L^{2} (T)$ with $g = f^{'}$ in the sense of Periodic L2 weak derivative on the circle.

Proof

technique · direct

1.1

Take the smooth periodic test function $φ = e_{- k}$ in the defining identity. Since $φ^{'} = - 2 π i k e_{- k}$ , it gives $- 2 π i k \hat{f} (k) = - \hat{g} (k)$ .

givenalgebra

2.1

Rearranging proves the formula; for $k = 0$ it says $\hat{g} (0) = 0$ , which is also the same test-function identity with $φ = 1$ .

step 1.1algebra∎

Depends on

Periodic L2 weak derivative on the circle

Used by

One ell-2 weak derivative implies an absolutely convergent Fourier series Corollary

Dependency tree · two levels

2 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

Michael E. Taylor, Fourier Analysis, Distributions, and Constant-Coefficient Linear PDE, Section 1 (standard reference, not scraped)