## 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.

### MSC:

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

### Keywords:

Walsh-Nörlund means; modulus of continuity
Full Text:

### References:

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