Notes on some points in the integral calculus. LI. (On \textit{Hilbert}'s double-series theorem, and some connected theorems concerning the convergence of infinite series and integrals.). (Q1470699)
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scientific article; zbMATH DE number 2610902
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Notes on some points in the integral calculus. LI. (On \textit{Hilbert}'s double-series theorem, and some connected theorems concerning the convergence of infinite series and integrals.). |
scientific article; zbMATH DE number 2610902 |
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Notes on some points in the integral calculus. LI. (On \textit{Hilbert}'s double-series theorem, and some connected theorems concerning the convergence of infinite series and integrals.). (English)
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1918
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Die Note bringt einen einfachen Beweis für den Satz: Ist \(\sum_{n=1}^\infty a_n^2\) konvergent, dann ist auch \[ \sum_{n=1}^\infty(\frac{a_1+a_2+\cdots+a_n}{2})^2 \] konvergent. Daraus folgt, wie schon früher (Messenger 45, 163; F. d. M. 45, 1290 (JFM 45.1290.*), 1914-15) angedeutet wurde, auch ein neuer Beweis des \textit{Hilbert}schen Satzes über die Beschränktheit der quadratischen Form \(\sum_{m,n=1}^\infty\frac{a_ma_n}{m+n}\). Es werden auch analoge Sätze über Integrale hergeleitet und die Beziehung zu den \textit{Fourier}schen Reihen kurz erwähnt.
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