Fourier series for singular measures and the Kaczmarz algorithm
نویسندگان
چکیده
Using the Kaczmarz algorithm, we obtain a Fourier series formulation for functions in the L2 space of singular measures on the unit circle. This formula is applied to the problem of finding reproducing kernel Hilbert spaces inside the classical Hardy space, where the norm is instead that of boundary integration with respect to a singular measure. We also give some conditions ensuring that these subspaces bear some essential semblances to the classical Hardy space.
منابع مشابه
Fourier Series for Singular Measures
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 in a sense μ-bandlimited.
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