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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 $\left(k+1\right)$-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 (generalized functions) 46F12 Integral transforms in distribution spaces 40G05 Cesàro, Euler, Nörlund and Hausdorff methods