## 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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