TheoremStatement: Literature-sourcedProof: AI-adaptedjudge 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.

The Wiener algebra is a unital commutative Banach algebra

Statement

Assume the Axiom of Countable Choice. Under pointwise operations on the continuous representatives, $A (T)$ is a commutative unital Banach algebra. Its unit is $e_{0} = 1$ , and $∥ f g ∥_{A} \leq ∥ f ∥_{A} ∥ g ∥_{A} .$

Facts & Assumptions

Given: The Axiom of Countable Choice, functions $f, g \in A (T)$ , and their absolutely summable coefficient sequences.

[L1]

Every $ℓ^{1}$ coefficient sequence has the continuous uniform synthesis stated in Absolutely summable Fourier coefficients give uniform convergence.

Proof

technique · direct

1.1

For finite Fourier sums, multiplying and collecting equal frequencies gives $\hat{f g} (n) = \sum_{k} \hat{f} (k) \hat{g} (n - k)$ .

givenalgebra

2.1

Truncate both coefficient series. By [L1] the truncations converge uniformly, and their convolution coefficients converge in $ℓ^{1}$ because $ℓ^{1} * ℓ^{1} \subseteq ℓ^{1}$ ; thus the formula in step 1.1 holds for $f, g$ .

L1step 1.1algebra

3.1

Tonelli's theorem for the nonnegative double series gives $∥ f g ∥_{A} \leq \sum_{n, k} ∣ \hat{f} (k) ∣ ∣ \hat{g} (n - k) ∣ = ∥ f ∥_{A} ∥ g ∥_{A} .$

step 2.1algebra

4.1

The coefficient map is an isometric bijection from $A (T)$ to $ℓ^{1} (Z)$ by [L1]; completeness, commutativity, and the unit therefore follow from those of $ℓ^{1}$ and $δ_{0}$ .

L1step 3.1algebra∎

Depends on

Used by

Dependency tree · two levels

4 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, Definition 4.1 and Theorem 4.2 (standard reference, not scraped)