Abstract
An old problem on absolute convergence of the re-expansion in the sine (cosine) Fourier series of an absolutely convergent cosine (sine) Fourier series is extended to the non-periodic case (Fourier transforms). Necessary and sufficient conditions are given as relations between the Fourier transforms and their Hilbert transforms. Sufficient conditions for integrability of the Hilbert transform are obtained.
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Acknowledgments
The author’s attention to this problem was brought by R.M. Trigub. Thanks to him as well as to A. Iosevich and J.-P. Kahane for stimulating discussions.
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Communicated by Hans G. Feichtinger.
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Liflyand, E. On Fourier Re-Expansions. J Fourier Anal Appl 20, 934–946 (2014). https://doi.org/10.1007/s00041-014-9339-0
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DOI: https://doi.org/10.1007/s00041-014-9339-0