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The Fourier transform of a function of bounded variation: symmetry and asymmetry

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ASC - Approximation, sampling and compression in data science

New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. The main result is an asymptotic formula for the  cosine Fourier transform. Such relations have previously been known only for the sine Fourier transform. To prove the mentioned result, not only a different space is considered but also a new way of proving such theorems is applied. Interrelations of various function spaces are studied in this context. The obtained results are used for proving new estimates for the Fourier transform of a radial function and completely new results on the integrability  of trigonometric series.




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