A complete system of orthogonal step functions
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- by Huaien Li and David C. Torney
- Proc. Amer. Math. Soc. 132 (2004), 3491-3502
- DOI: https://doi.org/10.1090/S0002-9939-04-07511-2
- Published electronically: July 22, 2004
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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 $L^p(0,1); \; 1 < p \in \mathbb {R}.$References
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Bibliographic 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
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3491-3502
- MSC (2000): Primary 11A25, 42C10, 42C30
- DOI: https://doi.org/10.1090/S0002-9939-04-07511-2
- MathSciNet review: 2084069