Summability of generalized multiple Fourier series by the method of Abel (Q920333)

Let \(K=\{x\in R^ m:| x_ i| \leq \pi\), \(i=1,2,...,m\}\) and the \(2\pi\)-periodic function f(x), \(x\in K\) have a unique singularity point \(x_ 0\). If \(x_ 0=0\) then there is some non-negative decreasing function \(\alpha\) (u) on (0,T] such that \(\| x\|^{\alpha (\| x\|)}f(x)\in L(K).\) Let M(u) be an integer-valued non-negative function on (0,T] satisfying M(u)\(\geq \alpha (u)\), \(0<u\leq u_ 0<T\) and \[ \{e^{-int}\}_ M=e^{-int}-\sum^{M(\| t\|)- 1}_{k=0}\frac{(-int)^ k}{k!}, \] where \(n=(n_ 1,n_ 2,...,n_ m)\), \(t=(t_ 1,t_ 2,...,t_ m)\), \(nt=n_ 1t_ 1+n_ 2t_ 2+...+n_ mt_ m\). Denote \(c_ n=(2\pi)^{-m}\int_{K}f(t)\{e^{- int}\}_ Mdt,\) \(n\in Z^ m\). The multiple trigonometric series \(\sum c_ ne^{inx}\) is called the Fourier M-series for f(x). The sum \(\sum_{n}| r|^{| n|}a_ n\), \(0<r<1\) is the Abel's average for the (divergent) multiple sum \(\sum a_ n.\) The author proves that for an arbitrary function f on K having a unique non-summing singularity on K the Fourier M-series is summed up to f(x) as soon as M(u) increases sufficiently fast as \(u\to +0\).