Almost Everywhere Convergence of Arithmetic Means of Walsh--Fourier Partial Sums Along Subsequences
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913095559938048 |
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| author | Goginava, Ushangi |
| author_facet | Goginava, Ushangi |
| contents | Let $S_m f$ denote the $m$-th partial sum of the Walsh-Fourier series of $f \in L^1$. For an increasing sequence $a=(a(n))_{n \geq 1}$ of positive integers, consider the arithmetic means
$$ σ_N f:=\frac{1}{N} \sum_{n=1}^N S_{a(n)} f . $$
Gát proved in 2019 that $σ_N f \rightarrow f$ almost everywhere for every $f \in L^1$ under the growth condition
$$ a(n+1) \geq\left(1+\frac{1}{n^δ}\right) a(n), \quad 0<δ<\frac{1}{2} . $$
We show that the same conclusion remains valid throughout the full range $0<δ<1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05146 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Almost Everywhere Convergence of Arithmetic Means of Walsh--Fourier Partial Sums Along Subsequences Goginava, Ushangi Classical Analysis and ODEs Let $S_m f$ denote the $m$-th partial sum of the Walsh-Fourier series of $f \in L^1$. For an increasing sequence $a=(a(n))_{n \geq 1}$ of positive integers, consider the arithmetic means $$ σ_N f:=\frac{1}{N} \sum_{n=1}^N S_{a(n)} f . $$ Gát proved in 2019 that $σ_N f \rightarrow f$ almost everywhere for every $f \in L^1$ under the growth condition $$ a(n+1) \geq\left(1+\frac{1}{n^δ}\right) a(n), \quad 0<δ<\frac{1}{2} . $$ We show that the same conclusion remains valid throughout the full range $0<δ<1$. |
| title | Almost Everywhere Convergence of Arithmetic Means of Walsh--Fourier Partial Sums Along Subsequences |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2605.05146 |