Almost-everywhere Fourier reconstruction of L log L signals
Let $\mathbb T = \mathbb R/\mathbb Z$ with normalized Lebesgue measure, and let $f : \mathbb T \to \mathbb C$ be measurable with $$\int_{\mathbb T} |f(x)| \log\bigl(e + |f(x)|\bigr)\, dx < \infty.$$ Let $\widehat f(k) = \int_{\mathbb T} f(x) e^{-2\pi i k x}\, dx$ and let $S_N f(x) = \sum_{k=-N}^{N} \widehat f(k) e^{2\pi i k x}$ be the symmetric partial sums. Carleson proved that $S_N f \to f$ almost everywhere for $f \in L^2(\mathbb T)$, and Hunt extended this to $L^p(\mathbb T)$ for every $p > 1$. Kolmogorov constructed an integrable function whose Fourier series diverges almost everywhere. Antonov proved almost-everywhere convergence for every $f$ with $\int_{\mathbb T} |f| \log(e + |f|) \log\log\log(e^{e^e} + |f|)\, dx < \infty$.
Problem. Must $S_N f(x) \to f(x)$ for almost every $x$ as $N \to \infty$? The assertion concerns the full sequence of partial sums, without averaging or restriction to a sparse subsequence.
Source: The L log L conjecture, as Di Plinio (2014) calls it, stated there as a weak-type bound for the Carleson operator on L log L. Stated by AIM — Open Applied Problems as almost-everywhere convergence.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CA — Classical Analysis and ODEs
- Secondary
- FA
- Keywords
- AIM Open Applied Problems, almost everywhere convergence, Carleson operator, Fourier series
References
@article{diplinio2014,
author = {Di Plinio, Francesco},
title = {Weak-{$L^p$} bounds for the {C}arleson and {W}alsh--{C}arleson operators},
journal = {C. R. Math. Acad. Sci. Paris},
volume = {352},
number = {4},
year = {2014},
pages = {327--331},
doi = {10.1016/j.crma.2014.02.005},
eprint = {1312.0398},
archivePrefix = {arXiv},
}
@article{carleson1966,
author = {Carleson, Lennart},
title = {On convergence and growth of partial sums of {F}ourier series},
journal = {Acta Math.},
volume = {116},
year = {1966},
pages = {135--157},
doi = {10.1007/BF02392815},
}
@incollection{hunt1968,
author = {Hunt, Richard A.},
title = {On the convergence of {F}ourier series},
booktitle = {Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967)},
publisher = {Southern Illinois Univ. Press},
address = {Carbondale, Ill.},
year = {1968},
pages = {235--255},
}
@article{kolmogorov1923,
author = {Kolmogoroff, A.},
title = {Une s{\'e}rie de {F}ourier--{L}ebesgue divergente presque partout},
journal = {Fund. Math.},
volume = {4},
year = {1923},
pages = {324--328},
doi = {10.4064/fm-4-1-324-328},
}
@article{antonov1996,
author = {Antonov, N. Yu.},
title = {Convergence of {F}ourier series},
journal = {East J. Approx.},
volume = {2},
number = {2},
year = {1996},
pages = {187--196},
}
@misc{aim-fourier-l-log-l-pointwise-convergence,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Almost-everywhere Fourier reconstruction of L log L signals}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/448-fourier-l-log-l-pointwise-convergence.md},
note = {Entry 448 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove that the ordinary symmetric Fourier partial sums of every complex-valued function in $L\log L$ on the circle converge almost everywhere to the function along the full sequence. This establishes the classical $L\log L$ sufficiency conjecture.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00683, "Almost-everywhere Fourier reconstruction of L log L signals", clopen.solutions.