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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A complete system of orthogonal step functions


Authors: Huaien Li and David C. Torney
Journal: Proc. Amer. Math. Soc. 132 (2004), 3491-3502
MSC (2000): Primary 11A25, 42C10, 42C30
Published electronically: July 22, 2004
MathSciNet review: 2084069
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Abstract: We educe an orthonormal system of step functions for the interval $[0,1]$. This system contains the Rademacher functions, and it is distinct from the Paley-Walsh system: its step functions use the Möbius function in their definition. Functions have almost-everywhere convergent Fourier-series expansions if and only if they have almost-everywhere convergent step-function-series expansions (in terms of the members of the new orthonormal system). Thus, for instance, the new system and the Fourier system are both complete for $L^p(0,1); \; 1 < p \in \mathbb{R}.$


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Additional Information

Huaien Li
Affiliation: Department of Mathematics, University of Texas–Pan American, Edinburg, Texas 78539
Email: huaien_li@hotmail.com

David C. Torney
Affiliation: Los Alamos National Laboratory, Los Alamos, New Mexico 87545
Email: dtorney@earthlink.net

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07511-2
PII: S 0002-9939(04)07511-2
Keywords: Basis, Gram-Schmidt, M{\"o}bius inversion
Received by editor(s): December 2, 2002
Received by editor(s) in revised form: August 11, 2003
Published electronically: July 22, 2004
Additional Notes: This research was supported by the U.S.D.O.E. through its University of California contract W-7405-ENG-36; LAUR #02-1465.
Communicated by: David Sharp
Article copyright: © Copyright 2004 American Mathematical Society