A complete system of orthogonal step functions

Li, Huaien; Torney, David

doi:10.1090/S0002-9939-04-07511-2

A complete system of orthogonal step functions
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by Huaien Li and David C. Torney PDF

Proc. Amer. Math. Soc. 132 (2004), 3491-3502 Request permission

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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