Approximation of the Quadratic Partial Sums of Double Walsh–Kaczmarz–Fourier Series by Nörlund Means

Abstract

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( $$ \alpha $$ , p), where $$ \alpha $$ > 0 and 1 ≤ p ≤ ∞. For p = ∞, 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 M´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 Moricz and Rhoades [J. Approxim. Theory, 50, 341–358 (1987)].