Matches in SemOpenAlex for { <https://semopenalex.org/work/W2768081422> ?p ?o ?g. }
- W2768081422 endingPage "133" @default.
- W2768081422 startingPage "71" @default.
- W2768081422 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]." @default.
- W2768081422 created "2017-11-17" @default.
- W2768081422 creator A5014951818 @default.
- W2768081422 date "2017-01-01" @default.
- W2768081422 modified "2023-10-17" @default.
- W2768081422 title "One-Dimensional Fourier Transforms" @default.
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