Alphabeta MathHow 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.
The Wiener algebra of the circle
Definition
With the Fourier convention of Period-one Fourier coefficients, partial sums, and convolution on the torus, define Its Wiener norm is . Functions in this definition are initially -classes. Assuming the Axiom of Countable Choice, the later Absolutely summable Fourier coefficients give uniform convergence supplies their distinguished continuous representatives.
Depends on
Used by
- Wiener inversion needs nonvanishing Counterexample
- A trigonometric polynomial in the Wiener algebra Example
- An absolutely convergent Fourier series that is not twice continuously differentiable Example
- Absolutely summable Fourier coefficients give uniform convergence Lemma
- Bernstein's absolute-convergence theorem Theorem
- The Wiener algebra is a unital commutative Banach algebra Theorem
Dependency tree · one level
1 result within one dependency step 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, Chapter 4 (standard reference, not scraped)
- Loukas Grafakos, Classical Fourier Analysis, Definition 3.3.15 (standard reference, not scraped)