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Zbl 1224.42014
Vindas, Jasson; Estrada, Ricardo
On the order of summability of the Fourier inversion formula. (English)
[J] Anal. Theory Appl. 26, No. 1, 13-42 (2010). ISSN 1672-4070; ISSN 1573-8175/e

Summary: We show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and a certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that, if the distribution has a distributional point value of order $k$, then its Fourier series is e.v. Cesàro summable to the distributional point value of order $k+1$. Conversely, we also show that, if the pointwise Fourier inversion formula is e.v. Cesàro summable of order $k$, then the distribution is the $(k+1)$-th derivative of a locally integrable function, and the distribution has a distributional point value of order $k+2$. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.

MSC 2000:: *42A24 Summability of trigonometric series
42A38 Fourier type transforms, one variable
46F10 Operations with distributions (generalized functions)
46F12 Integral transforms in distribution spaces
40G05 Traditional summability methods

Keywords: Fourier inversion formula; tempered distribution; distributional point value; Cesàro summability

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