Approximation of the Quadratic Partial Sums of Double Walsh-Kaczmarz-Fourier Series by Norlund Means

We discuss the Norlund means of quadratic partial sums for theWalsh-Kaczmarz-Fourier series of a function in L (p) . The rate of approximation by these means is investigated, in particular, in Lip( , p), where > 0 and 1 aecurrency sign p aecurrency sign a. For p = a, the set L (p) turns into the collection of continuous functions C. Our main theorem states that the approximation behavior of these two-dimensional Walsh-Kaczmarz-Norlund means is as good as the approximation behavior of the one-dimensional Walsh and Walsh-Kaczmarz-Norlund means. Earlier, the results for one-dimensional Norlund means of the Walsh-Fourier series were obtained by MA ' oricz and Siddiqi [J. Approxim. Theory, 70, No. 3, 375-389 (1992)] and Fridli, Manchanda, and Siddiqi [Acta Sci. Math. (Szeged), 74, 593-608 (2008)]. For one-dimensionalWalsh-Kaczmarz-Norlund means, the corresponding results were obtained by the author [Georg. Math. J., 18, 147-162 (2011)]. The case of two-dimensional trigonometric systems was studied by Mricz and Rhoades [J. Approxim. Theory, 50, 341-358 (1987)].