Fourier Series for Singular Measures

@article{Herr2017FourierSF,
  title={Fourier Series for Singular Measures},
  author={John E. Herr and Eric S. Weber},
  journal={Axioms},
  year={2017},
  volume={6},
  pages={7}
}
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 cne. We show that the coefficients cn can be computed in terms of the quantities f̂ (n) = ∫ 1 0 f (x)e −2πinxdμ(x). We also demonstrate a Shannon-type sampling theorem for functions that are… CONTINUE READING