Approximation by Norlund means of quadratical partial sums of double Walsh - Kaczmarz - Fourier series - Nagy, K. (Ukrains’kyi Matematychnyi Zhurnal, 2016)
We discuss the Norlund means of quadratic partial sums of the Walsh – Kaczmarz – Fourier series of a function in $L_p$. We investigate the rate of approximation by this means, in particular, in $\text{Lip}(α , p)$, where $α > 0$ and $1 ≤ p ≤ ∞$. For $p = ∞$, by $L_p$, we mean $C$, i.e., the collection of continuous functions.Our main theorem states that the approximation behavior of this two-dimensional Walsh – Kaczmarz –Norlund means is as good as the approximation behavior of the one-dimensional Walsh– and Walsh – Kaczmarz –Norlund means. Earlier results for one-dimensional N¨orlund means of the Walsh – Fourier series was given by M´oricz and Siddiqi [J.Approxim. Theory. – 1992. – 70, № 3. – P. 375 – 389] and Fridli, Manchanda and Siddiqi [Acta Sci. Math. (Szeged). – 2008. – 74. – P. 593 – 608], for one-dimensional Walsh – Kaczmarz –N¨orlund means by the author [Georg. Math. J. –2011. – 18. – P. 147 – 162] and for two-dimensional trigonometric system by M´oricz and Rhoades [J. Approxim. Theory. –1987. – 50. – P. 341 – 358].