#### Title

On the order of summability of the Fourier inversion formula

#### Document Type

Article

#### Publication Date

3-1-2010

#### Abstract

In this article 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 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. © Editorial Board of Analysis in Theory & Applications and Springer-Verlag Berlin Heidelberg 2010.

#### Publication Source (Journal or Book title)

Analysis in Theory and Applications

#### First Page

13

#### Last Page

42

#### Recommended Citation

Vindas, J., & Estrada, R.
(2010). On the order of summability of the Fourier inversion formula.* Analysis in Theory and Applications**, 26* (1), 13-42.
https://doi.org/10.1007/s10496-010-0013-3