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The paley‐wiener theorem for the radon transform
Author(s) -
Lax Peter D.,
Phillips Ralph S.
Publication year - 1970
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160230311
Subject(s) - mathematics , convolution (computer science) , radon transform , danskin's theorem , class (philosophy) , mathematical analysis , regular polygon , function (biology) , mean value theorem (divided differences) , convolution theorem , shift theorem , pure mathematics , extreme point , brouwer fixed point theorem , picard–lindelöf theorem , fixed point theorem , fourier transform , combinatorics , fractional fourier transform , geometry , machine learning , artificial intelligence , evolutionary biology , artificial neural network , computer science , biology , fourier analysis
We prove that the support of a complex‐valued function f in ℝ k is contained in a convex set K if and only if the support of its Radon transform k ( s , ω) is, for each ω, contained in s ≦ S K (ω); here S K is the support function of the set K . This theorem is used to determine the propagation speeds of hyperbolic differential equations with constant coefficients, to prove the nonexistence of point spectrum for a certain class of partial differential operators, and to give a simple reduction of Lions' convolution theorem to the one‐dimensional convolution theorem of Titchmarsh.