Abstract
A fundamental theme in classical Fourier analysis relates smoothness properties of functions to the growth and/or integrability of their Fourier transform. By using a suitable class of \(L^{p}\)-multipliers, a rather general inequality controlling the size of Fourier transforms for large and small argument is obtained. As consequences, quantitative Riemann–Lebesgue estimates are obtained and an integrability result for the Fourier transform is developed extending ideas used by Titchmarsh in the one dimensional setting.
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Notes
Fourier inversion actually holds everywhere. The Fourier inversion integral defines a continuous function which is equal to \(f\) a.e. This remark also applies to Corollaries 3.5, 4.4, and 4.6.
The result is stated in a form appropriate for our use; the hypothesis on \(\psi \) can be weakened. It is a generalization of an asymptotic result for the Fourier cosine transform (endpoint case of \(\mu =-1/2\)) due to Titchmarsh [17, Theorem 126].
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The author is grateful to the referee for several suggestions improving the exposition in this paper.
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Communicated by Hans G. Feichtinger.
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Bray, W.O. Growth and Integrability of Fourier Transforms on Euclidean Space. J Fourier Anal Appl 20, 1234–1256 (2014). https://doi.org/10.1007/s00041-014-9354-1
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DOI: https://doi.org/10.1007/s00041-014-9354-1