$(C‎,‎\alpha)$-Summability with Varying-parameters of Quadratical Partial Sums of Double Fourier Series on the Group of $2$-Adic Integers

Document Type : Original Article

Author

Anteneh Tilahun Adimasu

Department of Mathematics, College of Natural Sciences, Wollo University, Ethiopia

10.22034/kjm.2025.527536.3565

Abstract

In this paper, we prove that the maximal operator of the $(C,\alpha_{n})$-means of the quadratical partial sums of double Fourier series on the group of $2$-adic integers is quasi-local, weak type $(L^{1}, L^{1})$ and of strong type $(L^{\infty}, L^{\infty}).$ Moreover, we prove almost everywhere convergence of $(C,\alpha_{n})$-means of integrable functions (i.e. $ \sigma_{n}^{\alpha_{n}}f\longrightarrow f \quad a.e.), \, $ for $f\in L^{1}(\mathbb{I}^{2}), \, \mathbb{I}^{2} $ is two dimensional group on the $2$-adic integers in which $ \alpha_{n}\in (0,1)$ and $\alpha_{n}$ is varying parameter.

Keywords

($C,\alpha$)-means

two-dimensional Fourier series

almost everywhere convergence

maximal operator

Subjects

• 42 Harmonic analysis on Euclidean spaces

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