A partial converse to the Riemann--Lebesgue lemma for Bessel--Fourier series of order zero
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929555089915904 |
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| author | Babb, Ryan L. Acosta |
| author_facet | Babb, Ryan L. Acosta |
| contents | It is known that the Bessel--Fourier coefficients $f_m$ of a function $f$ such that $\sqrt{x}f(x)$ is integrable over $[0,1]$ satisfy $f_m/\sqrt{m}\to 0$. We show a partial converse, namely that for $0\leq α<1/2$ and any non-negative $a_m\to 0$, there is a function $f$ such that $x^{α+1}f(x)$ is integrable and its Bessel--Fourier coefficients $f_m$ satisfy $m^{-α}f_m\geq a_m$ and $m^{-α}f_m\to 0$. We conjecture that the same should be true when $α=\frac{1}{2}$, and discuss some consequences of this conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A partial converse to the Riemann--Lebesgue lemma for Bessel--Fourier series of order zero Babb, Ryan L. Acosta Classical Analysis and ODEs 33C10 (Primary), 42A16, 41A30 (Secondary) It is known that the Bessel--Fourier coefficients $f_m$ of a function $f$ such that $\sqrt{x}f(x)$ is integrable over $[0,1]$ satisfy $f_m/\sqrt{m}\to 0$. We show a partial converse, namely that for $0\leq α<1/2$ and any non-negative $a_m\to 0$, there is a function $f$ such that $x^{α+1}f(x)$ is integrable and its Bessel--Fourier coefficients $f_m$ satisfy $m^{-α}f_m\geq a_m$ and $m^{-α}f_m\to 0$. We conjecture that the same should be true when $α=\frac{1}{2}$, and discuss some consequences of this conjecture. |
| title | A partial converse to the Riemann--Lebesgue lemma for Bessel--Fourier series of order zero |
| topic | Classical Analysis and ODEs 33C10 (Primary), 42A16, 41A30 (Secondary) |
| url | https://arxiv.org/abs/2410.17681 |