Approximation by Nörlund means of quadratical partial sums of double Walsh-Fourier series. (English) Zbl 1240.42133

The main theorem of the paper states that the approximation behavior of the two-dimensional Walsh-Nörlund means of Marcinkiewicz type is so good as the approximation behavior of the one-dimensional Walsh-Nörlund means. More precisely, let \[ \boldsymbol{t}^w_n(f, x^1,x^2) :=\frac{1}{Q_n}\sum_{k=1}^nq_{n-k}S_{k,k}^w(f, x^1,x^2) \] be the two dimensional Walsh-Nörlund means of Marcinkiewicz type, where \(Q_n=\sum_{k=1}^{n-1}q_k\).
Then if \(f\in L^p\) (\(1\leq p\leq\infty\)) and \(q_k\) is nondecreasing, we have \[ \| \boldsymbol{t}_n^w(f)-f\| _p\leq\frac{c}{Q_n}\sum_{l=0}^{A-1}q_{n-2^l}2^l\omega_p(2^{-l},f)+O(\omega_p(2^{-A},f)), \] and the same for nonincreasing \(q_k\) if we additionally suppose that \[ \frac{n}{Q_n^2}\sum_{k=1}^{n-1}q_k^2=O(1). \] Here, \(A\) is the order of \(n\), supposed to be positive, and \(\omega_p(\delta,f)\) is the modulus of continuity.
The author discusses also the asymptotic behaviour of the norm of \(\boldsymbol{t}_n^w(f)-f\) if \(f\) is in different Lipschitz classes.


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