Fourier coefficients of continuous functions with sparse spectrum
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866913098616537088 |
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| author | Kulikov, Aleksei Saucedo, Miquel Tikhonov, Sergey |
| author_facet | Kulikov, Aleksei Saucedo, Miquel Tikhonov, Sergey |
| contents | Let $(r_k)$ be an increasing sequence and $(w_k)$ a positive sequence. We study the following question: is it true that for every sequence $(a_k)$ satisfying $\sum_{k=0}^\infty |a_k|^2 w_k^2 < \infty$ there exists a function $f\in C(\mathbb{T})$ such that $\hat{f}(2^k) = a_k$ and $\hat{f}(n) = 0$ for $n\notin \cup_k [2^k-r_k,2^k+r_k]$? We show that this is possible if and only if $\sup_{k\in\mathbb{N}}\sum_{n=[\log_2 r_k]}^k w_k^{-2} < \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06025 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fourier coefficients of continuous functions with sparse spectrum Kulikov, Aleksei Saucedo, Miquel Tikhonov, Sergey Classical Analysis and ODEs 42A16, 42A20 (Primary), 46E30 (Secondary) Let $(r_k)$ be an increasing sequence and $(w_k)$ a positive sequence. We study the following question: is it true that for every sequence $(a_k)$ satisfying $\sum_{k=0}^\infty |a_k|^2 w_k^2 < \infty$ there exists a function $f\in C(\mathbb{T})$ such that $\hat{f}(2^k) = a_k$ and $\hat{f}(n) = 0$ for $n\notin \cup_k [2^k-r_k,2^k+r_k]$? We show that this is possible if and only if $\sup_{k\in\mathbb{N}}\sum_{n=[\log_2 r_k]}^k w_k^{-2} < \infty$. |
| title | Fourier coefficients of continuous functions with sparse spectrum |
| topic | Classical Analysis and ODEs 42A16, 42A20 (Primary), 46E30 (Secondary) |
| url | https://arxiv.org/abs/2605.06025 |