Multi-Dimensional fourier transforms

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

We study the theory of multi-dimensional Fourier transforms, namely, the inversion formula and the convergence of Fourier transforms. We formulate the analogous results to those of Sections 2.1 – 2.4 for higher dimensions. In the first section, we introduce the Fourier transform for functions and for tempered distributions and give the most important results. Since these proofs are very similar to those of the one-dimensional ones, we omit the proofs. In Section 4.3, we consider four types of Dirichlet integrals of multi-dimensional Fourier transforms, i.e. the cubic, triangular, circular and rectangular Dirichlet integrals. Using the analogous results for the partial sums of multi-dimensional Fourier series proved in Section 4.2, we show that the Dirichlet integrals converge in the Lp(ℝd) -norm to the function (1 < p < ∞). The multi-dimensional version of Carleson’s theorem is also verified.

Original languageEnglish
Title of host publicationApplied and Numerical Harmonic Analysis
PublisherSpringer International Publishing
Pages203-227
Number of pages25
Edition9783319568133
DOIs
Publication statusPublished - Jan 1 2017

Publication series

NameApplied and Numerical Harmonic Analysis
Number9783319568133
ISSN (Print)2296-5009
ISSN (Electronic)2296-5017

Fingerprint

Dirichlet Integral
Fourier transform
Fourier transforms
Tempered Distribution
Inversion Formula
Fourier series
Partial Sums
Higher Dimensions
Triangular
Converge
Norm
Theorem

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Weisz, F. (2017). Multi-Dimensional fourier transforms. In Applied and Numerical Harmonic Analysis (9783319568133 ed., pp. 203-227). (Applied and Numerical Harmonic Analysis; No. 9783319568133). Springer International Publishing. https://doi.org/10.1007/978-3-319-56814-0_4

Multi-Dimensional fourier transforms. / Weisz, F.

Applied and Numerical Harmonic Analysis. 9783319568133. ed. Springer International Publishing, 2017. p. 203-227 (Applied and Numerical Harmonic Analysis; No. 9783319568133).

Research output: Chapter in Book/Report/Conference proceedingChapter

Weisz, F 2017, Multi-Dimensional fourier transforms. in Applied and Numerical Harmonic Analysis. 9783319568133 edn, Applied and Numerical Harmonic Analysis, no. 9783319568133, Springer International Publishing, pp. 203-227. https://doi.org/10.1007/978-3-319-56814-0_4
Weisz F. Multi-Dimensional fourier transforms. In Applied and Numerical Harmonic Analysis. 9783319568133 ed. Springer International Publishing. 2017. p. 203-227. (Applied and Numerical Harmonic Analysis; 9783319568133). https://doi.org/10.1007/978-3-319-56814-0_4
Weisz, F. / Multi-Dimensional fourier transforms. Applied and Numerical Harmonic Analysis. 9783319568133. ed. Springer International Publishing, 2017. pp. 203-227 (Applied and Numerical Harmonic Analysis; 9783319568133).
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