Criteria for absolute convergence of multiple Fourier series
Let \(m=(m_ 1,m_ 2,...,m_ n)\), where \(m_ 1,m_ 2,...,m_ n\) are integers, \(t=(t_ 1,t_ 2,...,t_ n)\in {\mathbb{R}}^ n\) and \(e^{imt}=e^{i(m_ 1t_ 1+m_ 2t_ 2+...+m_ nt_ n)}.\) Let \(\sum_{m}f_ me^{imt}\) be the Fourier series of a function f(t), integrable on \(T^ n=\{t:0\leq t_ k\leq 2\pi;k=1,2,...,n\}\) and \(2\pi\) periodic in each variable. We denote \(\| f\|_ A=\sum_{m}| f_ m|\) and \(\| f_2\| = \| f\|_{L_2(T^ n)}\). If \(\frac{\partial^ qf}{\partial t^ q\!_ k}\in L_ 2(T^ n)\) for some \(q=0,1,2,..\). (as usual, \(\partial^ 0f/\partial t^ 0\!_ k=f)\) then we put \(\omega^{(q)}\!\!\!_{j,k}(f,y)=\| \frac{\partial^ qf}{\partial t^ q\!_ k}(t_ 1,t_ 2,...,t_ j+y,...,t_ n)- \frac{\partial^ qf}{\partial t^ q\!_ k}(t_ 1,...,t_ n)\|_ 2.\) The main theorem proved by the author for the absolute sums of Fourier series reads as follows. Theorem: Let f(t) be a periodic function such that \((a)\quad \partial^ jf/\partial t^ j\!_ k, k=1,2,...,n\); \(j=0,1,...,q-1\); \(q=[n/2]\) are integrable functions, essentially absolutely continuous in \(t_ k\) (if \(n=1\), then (a) should be dropped), \((b)\quad \partial^ qf/\partial t^ q\!_ k\in L_ 2(T^ n)\) for \(k=1,2...,n\). Let \(j_ 1,j_ 2,...,j_ n\) be positive integers not larger than n. If n is even, then for some \(c=c(n)\), we have \[ (1)\quad \| f\|_ A\leq | f_{0,...,0}| +c\sum^{n}_{k=1}| \quad \| \frac{\partial^ qf}{\partial t^ q\!_ k}\|_ 2+\int^{\frac{1}{2}}_{0}\frac{\omega^{(q)}\!\!\!_{j_ k,k}(f,y)}{y| \log y|^{\frac{1}{2}}}dy|. \] Moreover,let j'\({}_ 1,j'\!_ 2,...,j'\!_ n\) and j''\({}_ 1,j''\!_ 2,...,j''\!_ n\) be positive integers not larger than n and such that each pair j'\({}_ k,j''\!_ k\) satisfies one of the conditions: j'\({}_ k=j''\!_ k=k\) or j'\({}_ k\neq j''\!_ k\). If n is odd, then for some \(c=c(n)\), we have \[ (2)\quad \| f\|_ A\leq | f_{0,0,...,0}| +c\sum^{n}_{k=1}[\| \frac{\partial^ qf}{\partial t^ q\!_ k}\|_ 2+\int^{\frac{1}{2}}_{0}\frac{\omega_{j'\!_ k,k}(f,y)+\omega_{j''\!_ k,k}(f,y)}{y^{3/2}}dy]. \] If we choose j'\({}_ k=j''\!_ k=k\), then (2) takes the form \[ \| f\|_ A\leq | f_{0,0,...,0}| +c\sum^{n}_{k=1}[\| \frac{\partial^ qf}{\partial t^ q\!_ k}\|_ 2+\int^{\frac{1}{2}}_{0}\frac{\omega^{(q)}\!\!\!_{k,k}(f,y)}{y^{ 3/2}}dy]. \] For \(n=1\), when \(q=0\), we obtain Bernstein's Theorem.
scientific article; zbMATH DE number 3849811