On the order of summability of the Fourier inversion formula. (English) Zbl 1224.42014

Vindas, Jasson; Estrada, Ricardo

Anal. Theory Appl. 26, No. 1, 13-42 (2010).

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:

42A24	Summability and absolute summability of Fourier and trigonometric series
42A38	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46F10	Operations with distributions and generalized functions
46F12	Integral transforms in distribution spaces
40G05	Cesàro, Euler, Nörlund and Hausdorff methods

Keywords:

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

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