Note on a theorem of Hilbert.
Note on a theorem of Hilbert.
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DOI：
10.1007/bf01199965
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发表时间：
1920-01-01
影响因子：
0.8
通讯作者：
Hardy, GH
中科院分区：
文献类型：
--
作者：
Hardy, GH
1. Hilbert在研究积分方程理论的过程中，证明了级数aoq-n是vonvergen，只要aoq-n收敛。这个定理是双正项级数理论中最简单、最美丽的定理之一，至少有五种本质上不同的证明已经发表。希尔伯特自己的证明依赖于傅立叶级数理论，魏尔在他的就职论文中概述了这一证明。维纳给出了另一个证明，舒尔又给出了两个证明;但是这些证明都不是所希望的那样简单的反初等的。因此，施库尔的第一个证明取决于理论的二次和biline~ r形式在一个inigy的变量;和他的seeon&（这无疑是最优雅的所有）的变化变量的双重积分。维纳的证明虽然是基本的，但显然是人为的。在这四个证明之外，我最近又加上了第五个，它似乎在简洁性上毫不欠缺。我首先注意到希尔伯特定理是另一个定理的直接推论，而这个定理本身似乎有些意思。这个定理如下：i）H. Weyl，，~ Singul~，re Integralgleichungen”，GS t~ en 1908，p. 88.- 2)FW Wiener，~ Elemen~ arer Beweis eines R~ ensa~ es yon Herrn Hilbert”，Math. Annalen，88（19i0），I？第861--866页。I.舒尔，Bemerkungen zur Theorie der贝施~ nkten Bi~ earformen m~ t unendlieh vielen Ver~ nderlichen”，Jo，u~ nal fiiz Math.，1~ 0（1912），pp. 1~ 28。4)GH哈代，，笔记中的一些积分oaleulus点（51），数学信使，48（1918），页，107--！12.
1. It was proved by Hilbert, in the course of his in~ s~ g~ tions in the theory of integral equations, that the series ao~ n q-n is vonvergen $ whenever~ am is co~ vergent. Of this theorem, whioh i~ one of the simplest and moat beautiful in the theory of double series of positive terms, at least five essentially different proofs have been published. Hilbert's own proof, which depends upon the theory of Fourier's series, is outlined by Weyl in his Inaugural-DissertationS). Another proof was given by WienerS), and two more by Schur~); but none of these proofs is as simple anti elementary as might be desired. Thus S chur's first proof depends upon the theory of quadratic and biline~ r forms in an in~ igy of variables; and his seeon& (which is unquestionably the most elegant of all) on a change of variables in a double integral. And Wiener's proof, while genuinely elementary, is distinctly artificial. To these four proofs I added recently 4) a fifth which seemed go me to lack nothing in simplicity. I observed first that Hilbert's theorem is an immediate corollaxy of another theorem wtiieh seems of Some interest in itself. This theorem is as follows: i) H. Weyl,,~ Singul~, re Integralgleichungen", GS t~ en 1908, p. 88.-2) FW Wiener,~ Elemen~ arer Beweis eines R~ ensa~ es yon Herrn Hilbert", Math. Annalen, 88 (19i0), I? P. 861--866.~) I. Schur,. Bemerkungen zur Theorie der besch~ nkten Bi~ earformen m~ t unendlieh vielen Ver~ nderlichen", Jo, u~ nal fiiz Math., 1~ 0 (1912), pp. 1~ 28. 4) GH Hardy,, Notes on some points in the integral oaleulus (51), Messenger of Mathematics, 48 (1918), pp, 107--! 12.