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Article

Fourier Series for Singular Measures

1
Department of Mathematics, Butler University, Indianapolis, IN 46208, USA
2
Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011, USA
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Author to whom correspondence should be addressed.
Academic Editors: Palle E.T. Jorgensen and Azita Mayeli
Received: 21 February 2017 / Revised: 23 March 2017 / Accepted: 24 March 2017 / Published: 28 March 2017
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [ 0 , 1 ) , every f L 2 ( μ ) possesses a Fourier series of the form f ( x ) = n = 0 c n e 2 π i n x . We show that the coefficients c n can be computed in terms of the quantities f ^ ( n ) = 0 1 f ( x ) e 2 π i n x d μ ( x ) . We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ -bandlimited. View Full-Text
Keywords: Fourier series; singular measure; Kaczmarz algorithm Fourier series; singular measure; Kaczmarz algorithm
MDPI and ACS Style

Herr, J.E.; Weber, E.S. Fourier Series for Singular Measures. Axioms 2017, 6, 7. https://doi.org/10.3390/axioms6020007

AMA Style

Herr JE, Weber ES. Fourier Series for Singular Measures. Axioms. 2017; 6(2):7. https://doi.org/10.3390/axioms6020007

Chicago/Turabian Style

Herr, John E., and Eric S. Weber. 2017. "Fourier Series for Singular Measures" Axioms 6, no. 2: 7. https://doi.org/10.3390/axioms6020007

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