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Approximation by matrix means of double Fourier series to continuous functions in two variables. (English) Zbl 0953.42007
Let \(A= (a_{mnjk})\) be a doubly infinite matrix which is positive rectangular, that is, \(a_{mnjk}= 0\) for \(j> m\) or \(k> n\), \(a_{mnjk}> 0\) for each \(0\leq j\leq m\) and \(0\leq k\leq n\), and with double row sums \(1: \sum^m_{j=0} \sum^n_{k=0} a_{mnjk}= 1\) for each \(m,n\geq 0\). A double sequence \((s_{jk})\) is said to be summable by \(A\) if \(t_{mn}:= \sum^m_{j=0} \sum^n_{k=0} a_{mnjk} s_{jk}\) tends to a finite limit as \(m,n\to\infty\).
The authors study the rate of uniform approximation by the \(A\) means of the rectangular partial sums of double Fourier series of continuous functions, \(2\pi\)-periodic in each variable. The results are given in terms of the modulus of symmetric smoothness. The rate of uniform approximation to the conjugate functions is also discussed.

42B08 Summability in several variables
41A25 Rate of convergence, degree of approximation