Weak \(\ell^1\) estimates for the generalized discrete Hilbert transforms (Q2721586)
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scientific article; zbMATH DE number 1616257
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Weak \(\ell^1\) estimates for the generalized discrete Hilbert transforms |
scientific article; zbMATH DE number 1616257 |
Statements
16 May 2002
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discrete Hilbert transform
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Cauchy integral
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atomic measure
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Weak \(\ell^1\) estimates for the generalized discrete Hilbert transforms (English)
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Let \(\{x_n\}^\infty_{-\infty}\subset {\mathbf{R}}^1\) (\(x_m\not=x_n\) for \(m\not=n\)) be fixed. The generalized discrete Hilbert transform \(H=H_{\{x_n\}}\) is defined by NEWLINE\[NEWLINE(H\alpha)(n)=\sum_{\ell\not=n}{\alpha(l)\over{x_n-x_l}}.NEWLINE\]NEWLINE The author proves that if \(H\) is bounded on \(\ell^2\), then it is also bounded from \(\ell^1\) to weak \(\ell^1\).
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