Analysis in Theory and Applications

, Volume 26, Issue 1, pp 13–42 | Cite as

On the order of summability of the Fourier inversion formula

Article

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.

Key words

Fourier inversion formula tempered distribution distributional point value Cesàro summability of Fourier series and integrals summability of distributional evaluations 

AMS (2010) subject classification

42A24 42A38 46F10 40G05 

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Copyright information

© Editorial Board of Analysis in Theory and Applications and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of Pure Mathematics & Computer AlgebraGhent UniversityGentBelgium
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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