DefinitionDefinition: Literature-sourcedProof: Not applicableverified 2026-09-23 (gpt-6-sol)

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Periodic L2 weak derivative on the circle

Definition

Assume the Axiom of Countable Choice for the torus Lebesgue integrals used below. For $f, g \in L^{2} (T)$ , say that $g$ is the periodic weak derivative of $f$ , written $g = f^{'}$ , if $\int_{0}^{1} f (x) φ^{'} (x) d x = - \int_{0}^{1} g (x) φ (x) d x$ for every smooth one-periodic complex-valued test function $φ$ . This is a statement about almost-everywhere classes, using the circle and integral convention of Period-one Fourier coefficients, partial sums, and convolution on the torus.

Depends on

Period-one Fourier coefficients, partial sums, and convolution on the torus

Used by

Dependency tree · two levels

8 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)