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One-Dimensional Fourier Transforms

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Book cover Convergence and Summability of Fourier Transforms and Hardy Spaces

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Abstract

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 \(L_{2}(\mathbb{R})\), \(L_{1}(\mathbb{R})\), \(L_{p}(\mathbb{R})\ (1 \leq p \leq 2)\) 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 \(L_{p}(\mathbb{R})\)-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].

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  1. Department of Numerical Analysis, Eötvös Loránd University, Budapest, Hungary

    Ferenc Weisz

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Weisz, F. (2017). One-Dimensional Fourier Transforms. In: Convergence and Summability of Fourier Transforms and Hardy Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-56814-0_2

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