A complete system of orthogonal step functions

We educe an orthonormal system of step functions for the interval [0,1][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 expa...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2004-12, Vol.132 (12), p.3491-3502
Main Authors: Li, Huaien, Torney, David C.
Format: Article
Language:English
Subjects:
Algebra
Exact sciences and technology
Fourier analysis
Inner products
Integers
Mathematical analysis
Mathematical completeness
Mathematical functions
Mathematical intervals
Mathematics
Number theory
Orthogonal functions
Orthogonality
Partial sums
Research article
Sciences and techniques of general use
Sine function
Step functions
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Summary:We educe an orthonormal system of step functions for the interval [0,1][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 Lp(0,1);1>p∈R.L^p(0,1); \; 1 > p \in \mathbb {R}.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-04-07511-2