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

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
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