Fourier Series for Singular Measures

2015/03/16 by Herr, John E., Weber, Eric S. · 3 citations
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1503.04856

Abstract

Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [0,1), every f∈ L²(μ) possesses a Fourier series of the form f(x)=∑_n=0^∞c_ne^2πinx. We show that the coefficients c_n can be computed in terms of the quantities f(n) = ∫₀¹ f(x) e^{-2πi n x} d μ(x). We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ-bandlimited.

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