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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