The Fourier transform in Lebesgue spaces

Erik Talvila

Czechoslovak Mathematical Journal (2025)

  • Issue: 1, page 179-191
  • ISSN: 0011-4642

Abstract

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For each f ∈ L p ( ℝ ) ( 1 ≤ p < ∞ ) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each p , a norm is defined so that the space of Fourier transforms is isometrically isomorphic to L p ( ℝ ) . There is an exchange theorem and inversion in norm.

How to cite

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Talvila, Erik. "The Fourier transform in Lebesgue spaces." Czechoslovak Mathematical Journal (2025): 179-191. <http://eudml.org/doc/299908>.

@article{Talvila2025,
abstract = {For each $f\in L^p(\{\mathbb \{R\})\}$ ($1\le p<\infty $) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each $p$, a norm is defined so that the space of Fourier transforms is isometrically isomorphic to $L^p(\{\mathbb \{R\})\}$. There is an exchange theorem and inversion in norm.},
author = {Talvila, Erik},
journal = {Czechoslovak Mathematical Journal},
keywords = {Fourier transform; Lebesgue space; tempered distribution; generalised function; Banach space; continuous primitive integral},
language = {eng},
number = {1},
pages = {179-191},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Fourier transform in Lebesgue spaces},
url = {http://eudml.org/doc/299908},
year = {2025},
}

TY - JOUR
AU - Talvila, Erik
TI - The Fourier transform in Lebesgue spaces
JO - Czechoslovak Mathematical Journal
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 179
EP - 191
AB - For each $f\in L^p({\mathbb {R})}$ ($1\le p<\infty $) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each $p$, a norm is defined so that the space of Fourier transforms is isometrically isomorphic to $L^p({\mathbb {R})}$. There is an exchange theorem and inversion in norm.
LA - eng
KW - Fourier transform; Lebesgue space; tempered distribution; generalised function; Banach space; continuous primitive integral
UR - http://eudml.org/doc/299908
ER -

References

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  12. Stein, E. M., Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series 32. Princeton University Press, Princeton (1971). (1971) Zbl0232.42007MR0304972
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  14. Talvila, E., 10.1080/00029890.2019.1632629, Am. Math. Mon. 126 (2019), 717-727. (2019) Zbl1422.42007MR4009888DOI10.1080/00029890.2019.1632629
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