Jigneshkumar Bachubhai Shingod[a] and Bhikha Lila Ghodadra[b]
On the \( L^r \) convergence of Vilenkin series for \( 0 < r \le 1 \)
Pages: 459-488
Received: 5 October 2023
Accepted in revised form: 14 February 2024
Mathematics Subject Classification: 42C10, 40A05, 42A20, 40E05.
Keywords: Multiplicative systems, bounded Vilenkin system, Sequences of bounded variation, de la Vallée Poussin mean, Tauberian condition.
Authors address:
[a],[b]: The Maharaja Sayajirao University of Baroda, Faculty of Science, Department of Mathematics, Vadodara, 390 002, India
Abstract: In this paper we study the pointwise and uniform convergence, and convergence in the pseudo metric of \( L^r(0,1) \), for \( 0 < r < 1 \), of orthogonal series with respect to multiplicative systems with coefficients of generalized bounded variation. We also study the \( L^1 \)-convergence with some Tauberian conditions.