Integrated orthonormal series

James R. McLaughlin

Vol. 42 (1972), No. 2, 469–475

DOI: 10.2140/pjm.1972.42.469

Abstract

Throughout this paper the author defines

where 0 < α ≦ 2,a ≦ t ≦ b, and {φ_m} is a sequence in L¹[a,b], usually orthonormal. In this paper, F_α(t) is studied for the Haar, Walsh, trigonometric, and general orthonormal sequences. For instance, it is proved that for the Haar system F_α(t) satisfies a Lipschitz condition of order α∕2 in [0,1] and that this result is best possible for any complete orthonormal sequence. An application is also given regarding the absolute convergence of Walsh series.

Mathematical Subject Classification

Primary: 42A60

Secondary: 42A28

Milestones

Received: 25 February 1971

Revised: 24 April 1972

Published: 1 August 1972