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Matrix Summability of Walsh–Fourier Series

Author

Listed:
  • Ushangi Goginava

    (Department of Mathematical Sciences, United Arab Emirates University, Al Ain P.O. Box 15551, United Arab Emirates)

  • Károly Nagy

    (Institute of Mathematics and Computer Sciences, Eszterházy Károly Catholic University, Leányka Street 4, H3300 Eger, Hungary)

Abstract

The presented paper discusses the matrix summability of the Walsh–Fourier series. In particular, we discuss the convergence of matrix transforms in L 1 space and in C W space in terms of modulus of continuity and matrix transform variation. Moreover, we show the sharpness of our result. We also discuss some properties of the maximal operator t ∗ ( f ) of the matrix transform of the Walsh–Fourier series. As a consequence, we obtain the sufficient condition so that the matrix transforms t n ( f ) of the Walsh–Fourier series are convergent almost everywhere to the function f . The problems listed above are related to the corresponding Lebesgue constant of the matrix transformations. The paper sets out two-sides estimates for Lebesgue constants. The proven theorems can be used in the case of a variety of summability methods. Specifically, the proven theorems are used in the case of Cesàro means with varying parameters.

Suggested Citation

  • Ushangi Goginava & Károly Nagy, 2022. "Matrix Summability of Walsh–Fourier Series," Mathematics, MDPI, vol. 10(14), pages 1-25, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:14:p:2458-:d:862816
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