Paley-Wiener-Type Theorems Associated To Laplace-Bessel Operator

It is well known that Fourier-Bessel transform and its related Laplace-Bessel operator are important for solving the axisymmetric problem in high dimensional space.At present,someone has proposed a method for extracting the micro motion frequency of radar targets in discrete signal processing;in harmonic analysis,Y.Othmani and K.Trimeche have studied some properties and characterized the image of C∞-function under Fourier-Bessel transform.Based on their research,this thesis extends their properties and real PaleyWiener theorem to LP space for further development.First of all,since the expression of the Fourier-Bessel transform is the mixture of the classical Fourier transform and Hankel transform,we can get the Parseval equation of the Fourier-Bessel transformation,Plancherel theorem and so on by referring to their research methods.Then Hausdorff-Young inequality can be obtained by using the RieszThorin theorem.Next,to obtain the real Paley-Wiener theorem for the Laplace-Bessel operator on Lγp(Rn)for the Laplace-Bessel operator with different compact supports,the HausdorffYoung inequality is used to classify the cases of 1≤p<2 and 2≤p<∞.This proof method is more concise than that of Y.Othmani and K.Trimeche,and can extend the real Paley-Wiener theorem to Lγp(Rn)(1≤p≤∞).Finally,for 1≤p≤∞ and a fixed compact set K(?)Rn,we have obtained about Laplace-Bessel operator’s sufficient and necessary conditions for the norm sequences on Lp(Rn)space such that their spectrum are included on K.