One-Dimensional fourier transforms

Research output: Chapter in Book/Report/Conference proceedingChapter


In this chapter, we study the theory of one-dimensional Fourier transforms, the inversion formula, convergence and summability of Fourier transforms. In the first two sections, we introduce the Fourier transform for Schwartz functions and we extend it to L2(ℝ), L1(ℝ), Lp(ℝ)(1≤p≤2) functions as well as to tempered distributions. We prove some elementary properties and the inversion formula. In Sect. 2.4, we deal with the convergence of Dirichlet integrals. Using some results for the partial sums of Fourier series proved in Sect. 2.3, we show that the Dirichlet integrals converge in the Lp(ℝ) -norm to the function (1 < p < ∞). The proof of Carleson’s theorem, i.e. that of the almost everywhere convergence can be found in Carleson [52], Grafakos [152], Arias de Reyna [8], Muscalu and Schlag [253], Lacey [207] or Demeter [88].

Original languageEnglish
Title of host publicationApplied and Numerical Harmonic Analysis
PublisherSpringer International Publishing
Number of pages63
Publication statusPublished - Jan 1 2017

Publication series

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


ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Weisz, F. (2017). One-Dimensional fourier transforms. In Applied and Numerical Harmonic Analysis (9783319568133 ed., pp. 71-133). (Applied and Numerical Harmonic Analysis; No. 9783319568133). Springer International Publishing.