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A complete system of orthogonal step functions
Author(s):
Huaien
Li;
David
C.
Torney
Journal:
Proc. Amer. Math. Soc.
132
(2004),
3491-3502.
MSC (2000):
Primary 11A25, 42C10, 42C30
Posted:
July 22, 2004
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Abstract:
We educe an orthonormal system of step functions for the interval ![$[0,1]$](/proc/2004-132-12/S0002-9939-04-07511-2/gif-abstract0/img1.gif) . 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 
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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:
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
Posted:
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 of article:
Copyright
2004,
American Mathematical Society
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