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Vol. 52 No. 2 (2026)
Serdica Mathematical Journal

Articles

On the convergence of double Legendre series with coefficients of generalized bounded variation

https://doi.org/10.55630/serdica.2026.52.157-184

Published 24-06-2026

  • Jigneshkumar B. Shingod
    • ,
  • Bhikha L. Ghodadra

Jigneshkumar B. Shingod
Government Arts and Science College, Dediapada and The Maharaja Sayajirao University of Baroda

Bhikha L. Ghodadra
The Maharaja Sayajirao University of Baroda

Abstract

We investigate the convergence of double Legendre series whose coefficients belong to classes of bounded and generalized bounded variation. Under these assumptions, we establish pointwise, uniform, and \(L^{r}\)-convergence for \(0<r<1\). In addition, we study Cesàro and de la Vallée Poussin summability and obtain Tauberian-type results linking summability to ordinary convergence. The results extend known one-dimensional convergence theorems to the setting of double orthogonal expansions.

2020 Mathematics Subject Classification:

42C10, 40G05, 40E05, 26A45, 41A10

Keywords

Double Legendre Fourier series, Generalized bounded variation, $L^r$-Convergence, Uniform Convergence, Tauberian theorem, Orthogonal series

Full text

Author Details

Jigneshkumar B. Shingod

Department of Mathematics
Faculty of Science
The Maharaja Sayajirao University of Baroda
Vadodara – 390 002, Gujarat, India
and
Department of Mathematics
Government Arts and Science College
Dediapada, Narmada – 393040, Gujarat, India

e-mail: jbshingod1010@gmail.com

Bhikha L. Ghodadra

Department of Mathematics
Faculty of Science
The Maharaja Sayajirao University of Baroda
Vadodara – 390 002, Gujarat, India

e-mail: bhikhu_ghodadra@yahoo.com

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