Approximation by Nörlund means of double Walsh-Fourier series for Lipschitz functions. (English) Zbl 1243.42038

The Nörlund means of rectangular partial sums of a double Walsh-Fourier series are defined by \[ T_{m,n}\left( f;x^{1},x^{2}\right) :=\frac{1}{Q_{m,n}}\sum_{j=0}^{m} \sum_{k=0}^{n}q_{m-j,n-k}S_{j,k}\left( f;x^{1},x^{2}\right) , \] where \[ Q_{m,n}:=\sum_{j=0}^{m}\sum_{k=0}^{n}q_{j,k}. \] The main aim of this paper is to investigate the rate of approximation by the Nörlund means \(T_{m,n}\left( f;x^{1},x^{2}\right) \) the double Walsh-Fourier series of a function in \(L_{p},1\leq p\leq \infty \).
Earlier results on one-dimensional Nörlund means of the Walsh-Fourier series were given by F. Móricz and A. H. Siddiqi [J. Approximation Theory 70, No. 3, 375–389 (1992; Zbl 0757.42009)].


42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42B08 Summability in several variables


Zbl 0757.42009
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