On summability of subsequences of partial sums of trigonometric Fourier series (Q1261256)

\textit{Ya. S. Bugrov} [Izdatel'stvo Saratov. Univ. 3-13 (1986)] obtained an estimate for the deviation of \(f\in L^ p[-\pi,\pi]\) \((1\leq p\leq +\infty)\) from the linear means, induced by general regular matrices, of the partial sums of its Fourier series in \(L_ p\)-norm and extended this result to multiple Fourier series. In this paper, the author has obtained estimates for the deviation of \(f\in L^ p[-\pi,\pi]^ n\) \((1\leq p\leq +\infty\) and \(n\geq 1)\) from the multiple arithmetic means of the sequence of dyadic rectangular partial sums of \(n\)-multiple trigonometric Fourier series of \(f\in L^ p[-\pi,\pi]^ n\) in the norm of the corresponding space. Lastly, he has shown that the results so obtained, in the case when \(f\in C[-\pi,\pi]^ n\) \((n\geq 2)\), are best possible in a certain sense. For the sake of simplicity, the author has given a proof for the space \(C[-\pi,\pi]^ 2\). The proof for \(L[-\pi,\pi]^ 2\), being simple, has been omitted.