Quasiunconditional basis property of the Faber – Schauder system
We prove that, for any $0 < \delta < 1$, there exists a measurable set $E_{\delta} \subset [0, 1], \mathrm{m}\mathrm{e}\mathrm{s} (E_{\delta }) > 1 \delta $, such that for any function $f \in C[0, 1]$, one can find a function $\widetilde f \in C[0, 1]$ that coincides with...
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| Datum: | 2019 |
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| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Russisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2019
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1432 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | We prove that, for any $0 < \delta < 1$, there exists a measurable set $E_{\delta} \subset [0, 1], \mathrm{m}\mathrm{e}\mathrm{s} (E_{\delta }) > 1 \delta $, such that for any function
$f \in C[0, 1]$, one can find a function $\widetilde f \in C[0, 1]$ that coincides with f on E\delta , and the Fourier – Faber – Schauder series
for the function $\widetilde f$ unconditionally converges in $C[0, 1]$. Moreover, the moduli of the nonzero Fourier – Faber – Schauder
coefficients of the function $\widetilde f$ coincide with the elements of a given sequence $\{ b_n\}$ satisfying the condition
$$b_n \downarrow 0,\; \sum^{\infty }_{n=1} frac{b_n}{n} = +\infty .$$ |
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