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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.

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