Abstract
We obtain sufficient conditions for the absolute convergence of Fourier series for functions of L 2dψ depending on the properties of the function being expanded and the rate of growth of the sums\(\sum\nolimits_{k = 1}^n \varphi _k^2 (x)\) of the system of functions {ϕk(t)} orthonormalized in [a, b] with respect to dψ(t). We show that if at some point xε [a, b] the function ψ(t) has a discontinuity, at that point the Fourier series of any functionf(t) ε L 2dψ , converges absolutely.
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Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 511–516, November, 1972.
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Zinov'ev, A.S. The absolute convergence of orthogonal series. Mathematical Notes of the Academy of Sciences of the USSR 12, 743–746 (1972). https://doi.org/10.1007/BF01099056
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DOI: https://doi.org/10.1007/BF01099056