Abstract
The article studies the convergence of trigonometric Fourier series via a new Tauberian theorem for Cesàro summable series in abstract normed spaces. This theorem generalizes some known results of Hardy and Littlewood for number series. We find sufficient conditions for the convergence of trigonometric Fourier series in homogeneous Banach spaces over the circle. These conditions are expressed in terms of the Fourier coefficients and are weaker than Hardy’s condition. We give a description of all Banach function spaces given over the circle and endowed with a norm been equivalent to a norm in a homogeneous Banach space. We study interpolation properties of such spaces and give new examples of them. We extend the classical Fejér theorem on the uniform Cesàro summability of the Fourier series on sets by means of a refined version of Cantor’s theorem on the uniform continuity of a mapping between metric spaces. We also generalize the classical Hardy theorem on the uniform convergence of the Fourier series on sets.
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Funding
This work was funded by the Czech Academy of Sciences within Grant RVO:67985840 and by the National Academy of Sciences of Ukraine. The authors was also supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant agreement No. 873071 (SOMPATY: Spectral Optimization: From Mathematics to Physics and Advanced Technology).
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Conceptualization by Vladimir Mikhailets; Sects. 1 and 4 are performed by Vladimir Mikhailets and Aleksandr Murach; Sect. 2 by Vladimir Mikhailets and Oksana Tsyhanok; Sect. 3 by Aleksandr Murach; Sects. 5 and 6 by Vladimir Mikhailets. All authors read and approved the final manuscript.
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Mikhailets, V., Murach, A. & Tsyhanok, O. A New Look at Old Theorems of Fejér and Hardy. Results Math 79, 88 (2024). https://doi.org/10.1007/s00025-023-02114-y
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DOI: https://doi.org/10.1007/s00025-023-02114-y