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

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Ukrainian Mathematical Journal Aims and scope

We discuss the Nörlund 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–Nörlund means is as good as the approximation behavior of the one-dimensional Walsh and Walsh–Kaczmarz–Nörlund means.

Earlier, the results for one-dimensional Nörlund 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–Nörlund 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 Móricz and Rhoades [J. Approxim. Theory, 50, 341–358 (1987)].

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Authors and Affiliations

  1. Institute of Mathematics and Computer Sciences, College of Nyíregyháza, Nyíregyháza, Hungary

    K. Nagy

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 1, pp. 87–105, January, 2016.

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Nagy, K. Approximation of the Quadratic Partial Sums of Double Walsh–Kaczmarz–Fourier Series by Nörlund Means. Ukr Math J 68, 94–114 (2016). https://doi.org/10.1007/s11253-016-1211-8

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  • DOI: https://doi.org/10.1007/s11253-016-1211-8

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