On the localization of the Riesz means of multiple Fourier series of distributions
Let \(\mathbb T^N\) be the \(N\)-torus. Any distribution \(f\) in \(\varepsilon'(\mathbb T^N)\) has the Fourier series \[ f=(2\pi)^{-N/2} \sum_{n\in\mathbb{Z}^N} f_n\exp(inx), \] which converges in the weak topology. Consider the polynomial \[ P_m(n)= \Biggl(\sum^{r+1}_{j=1} n^2_j\Biggr)^{m+1}+ \Biggl(\sum^N_{j=r+2} n^2_j\Biggr)^m+ \Biggl(\sum^N_{j=1} n^2_j\Biggr) \] for \(n= (n_1,n_2,\dots, n_N)\in\mathbb{Z}^N\). The author studies the Riesz means of the Fourier series \[ E^s_\lambda f(x)= (2\pi)^{-N/2} \sum_{n\in\Lambda(\lambda)} \Biggl(1-{P_m(n)\over\lambda}\Biggr)^s\,f_n\exp(inx), \] where the set \(\Lambda(\lambda)\) is defined by \[ \Lambda(\lambda)= \{n\in\mathbb{Z}^N: P_m(n)< \lambda\}. \] The author obtains the followng localization theorem. Let \(f\in L^{-\alpha}_p\cap \varepsilon'(T^N)\), \(1< p\leq 2\), \(\alpha> 0\) and let \(f\) coincide with zero in \(\Omega\subset \mathbb T^N\). If \[ s>\max\Biggl\{{(N- r-1)(1-1/2m)\over p}+ r/2,{N-1\over 2}\Biggr\}+\alpha, \] then \(\lim_{\lambda\to\infty}\, E^s_\lambda f(x)= 0\) uniformly on any compact set \(K\subset\Omega\).
scientific article; zbMATH DE number 5997822