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On the order of summability of the Fourier inversion formula

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

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Chicago
Vindas Diaz, Jasson, and Ricardo Estrada. 2010. “On the Order of Summability of the Fourier Inversion Formula.” Analysis in Theory and Applications 26 (1): 13–42.
APA
Vindas Diaz, J., & Estrada, R. (2010). On the order of summability of the Fourier inversion formula. ANALYSIS IN THEORY AND APPLICATIONS, 26(1), 13–42.
Vancouver
1.
Vindas Diaz J, Estrada R. On the order of summability of the Fourier inversion formula. ANALYSIS IN THEORY AND APPLICATIONS. 2010;26(1):13–42.
MLA
Vindas Diaz, Jasson, and Ricardo Estrada. “On the Order of Summability of the Fourier Inversion Formula.” ANALYSIS IN THEORY AND APPLICATIONS 26.1 (2010): 13–42. Print.
@article{1260626,
  abstract     = {In this article we show that the order of the point value, in the sense of \unmatched{0141}ojasiewicz, 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. Ces{\`a}ro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Ces{\`a}ro 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.},
  author       = {Vindas Diaz, Jasson and Estrada, Ricardo},
  issn         = {1672-4070},
  journal      = {ANALYSIS IN THEORY AND APPLICATIONS},
  keyword      = {Ces{\`a}ro summability of Fourier series and integrals,distributional point value,tempered distribution,Fourier inversion formula,summability of distributional evaluations},
  language     = {eng},
  number       = {1},
  pages        = {13--42},
  title        = {On the order of summability of the Fourier inversion formula},
  url          = {http://dx.doi.org/10.1007/s10496-010-0013-3},
  volume       = {26},
  year         = {2010},
}

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