Estimates for the strong de la Vallée Poussin means in case of periodic continuous functions of two variables (Q1370579)

The authors evaluate the strong de la Vallée-Poussin averages formed with the help of a periodic bivariate function and the partial sums of its double Fourier series. Using the notations of the paper, the main result reads as follows: Let \(m,n,\mu,\nu\in \mathbb{N}_0\), \(m\leq n\) and \(\mu\leq\nu\). Then, for every \(p\in(0, \infty)\), \[ \begin{multlined}\Biggl|\Biggl\{{1\over (m+1)(\mu+ 1)} \sum^n_{k= n-m} \sum^\nu_{\ell= \nu-\mu}|S_{k,\ell} [f]- f|^p\Biggr\}^{1/p}\Biggr|_C\\ \leq K(p) E_{n- m,\nu-\mu}(f) \log{2(n+ 1)\over m+1} \log{2(\nu+1)\over \mu+1}.\end{multlined} \] Two more theorems are proved.