we-solve (beta)

Math-2397 · mathematics · Analysis

Almost-Everywhere Fourier Convergence in $L^{1}$

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Characterize the functions $f\in L^1(\mathbb T)$, for $\mathbb T=\mathbb R/\mathbb Z$, whose Fourier series converges almost everywhere.

Formulation references: [S1], [S2].

Short English statement

Characterize the functions $f\in L^1(\mathbb T)$, for $\mathbb T=\mathbb R/\mathbb Z$, whose Fourier series converges almost everywhere.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2397-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2397

Fourier convergence at the Orlicz endpoint: For a complex function on the circle satisfying the $L\log L$ integrability condition, the paper claims convergence of its symmetric Fourier sums almost everywhere. The limit is asserted for the ordinary full sequence of partial sums. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Almost-Everywhere Fourier Convergence in \$L^{1}\$

OpenAI's claim

Fourier convergence at the Orlicz endpoint: For a complex function on the circle satisfying the $L\log L$ integrability condition, the paper claims convergence of its symmetric Fourier sums almost everywhere. The limit is asserted for the ordinary full sequence of partial sums.

Mathematical statement of the claimed result

Mathematical claim 1.
On $\mathbb T=\mathbb R/(2\pi\mathbb Z)$ with normalized measure $\mu$, suppose $\int|f|\log(2+|f|)\,d\mu<\infty$. Set $\widehat f(k)=\int f(x)e^{-ikx}\,d\mu(x)$ and $S_Nf=\sum_{k=-N}^N\widehat f(k)e^{ikx}$. The claimed conclusion is \[S_Nf(x)\longrightarrow f(x)\quad\text{for almost every }x,\quad N=0,1,2,\ldots.\] It applies to complex-valued inputs and the full sequence of symmetric partial sums, under the stronger $L\log L$ hypothesis rather than merely $L^1$.

Outline of the argument

Fourier convergence at the Orlicz endpoint: A bounded nonnegative density is first controlled on the set where its spatial maximal average is moderate. The corresponding modulated dyadic sums are stopped on a finite binary tree, giving a good-set integral estimate uniform in the depth. Averaging delayed Haar kernels recovers the Hilbert kernel; doubly exponential height layers then make both the integral costs and exceptional-set measures summable.

Formal verification

Fourier convergence at the Orlicz endpoint: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

Go to contribution →