Integrability theorem of multiple trigonometric series
The aim of the paper under review is to extend a theorem by \textit{R. P. Boas} [Quart. J. Math., Oxford II. Ser. 3, 217-221 (1952; Zbl 0046.296)] from single to double cosine series as follows. Theorem 4. Let \(\{a_{mn}:\) \(m,n=1,2,...\}\) be a double sequence such that \((i)\quad a_{mn}\geq 0,\) (ii) \(\{a_{mn}\}\) is monotonically decreasing to zero in m for each fixed n, and in n for each fixed m; \[ (iii)\quad \Delta^{(2)}a_{mn}:=a_{mn}-a_{m+1,n}- a_{m,n+1}+a_{m+1,n+1}\geq 0; \] and let \[ f(x,y):=\sum^{\infty}_{m=1}\sum^{\infty}_{n=1}a_{mn} \cos mx \cos ny \] (the series converges everywhere, except possibly at \(x=0\) or \(y=0\) mod \(2\pi\)). Then for any \(0<\alpha,\beta <1,\) \(x^{-\alpha}y^{- \beta}f(x,y)\in L^ 1([0,2\pi]\times [0,\pi])\) if and only if \(\sum \sum m^{\alpha -1}n^{\beta -1}a_{mn}<\infty\).
scientific article; zbMATH DE number 4205090