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Journal of the London Mathematical Society 1999 59(1):252-262; doi:10.1112/S0024610798006875
© 1999 by London Mathematical Society
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© The London Mathematical Society

Harmonic and Logarithmic Summability of Orthogonal Series are Equivalent up to a Set of Measure Zero

F. Móricz and U. Stadtmüller

Bolyai Institute, University of Szeged Aradi vértanúk tere 1, 6720 Szeged, Hungary
Abt. Math. III, Universität Ulm 89069 Ulm, Germany

Received August 1996. Revision received 10 February 1997.

We prove Tauberian theorems from Jp-summability methods of power series type to ordinary convergence, respectively Mp-summability methods of weighted means. Particular cases are the Abel and Cesàro, as well as logarithmic and harmonic summability. Besides numerical series, we also consider orthogonal series with coefficients from L2. In the latter case, it turns out that one of our Tauberian conditions is satisfied almost everywhere on the underlying measure space, thereby proving the claim stated in the title.


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