Vol. 42, No. 2, 1972

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Integrated orthonormal series

James R. McLaughlin

Vol. 42 (1972), No. 2, 469–475
Abstract

Throughout this paper the author defines

       ∑∞       α   ∑∞   ∫ t        α
Fα(t) =   |Φm (t)| =     | a φm (x)dαj|
m=1          m=1

where 0 < α 2,a t b, and {φm} is a sequence in L1[a,b], usually orthonormal. In this paper, Fα(t) is studied for the Haar, Walsh, trigonometric, and general orthonormal sequences. For instance, it is proved that for the Haar system Fα(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
Authors
James R. McLaughlin