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Fourier inversion theorem
About: Fourier inversion theorem is a research topic. Over the lifetime, 2469 publications have been published within this topic receiving 67655 citations.
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TL;DR: In this article, a systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering, where the form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator.
Abstract: A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed The relationship of the scattering theory and Backlund transformations is brought out In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems
2,746 citations
Book•
01 Jan 1962
TL;DR: In this paper, the basic theorem of Fourier analysis on ordered groups has been studied, and the structure of locally compact Abelian groups has also been studied in the context of topology.
Abstract: The Basic Theorems of Fourier Analysis. The Structure of Locally Compact Abelian Groups. Idempotent Measures. Homomorphisms of Group Algebras. Measures and Fourier Transforms on Thin Sets. Functions of Fourier Transforms. Closed Ideals in L 1 (G). Fourier Analysis on Ordered Groups. Closed Subalgebras of L 1 (G). Appendices: Topology, Topological Groups, Banach Spaces, Banach Algebras, Measure Theory. Bibliography. List of Special Symbols. Index.
2,443 citations
Book•
01 Jan 1968TL;DR: In this article, the convergence of Fourier series on T and convergence of the conjugate function on T was studied, where T is the length of the line of a vector.
Abstract: 1. Fourier series on T 2. The convergence of Fourier series 3. The conjugate function 4. Interpolation of linear operators 5. Lacunary series and quasi-analytic classes 6. Fourier transforms on the line 7. Fourier analysis on locally compact Abelian groups 8. Commutative Banach algebras A. Vector-valued functions B. Probabilistic methods.
2,079 citations