On Eigenfunctions of the Fourier Transform

A nontrivial example of an eigenfunction in the sense of the theory of distributions for the planar Fourier transform was described by the authors in their previous work. In this paper, a method for obtaining other eigenfunctions is proposed. Positive homogeneous distributions in ℝ n of order − n /2...

Full description

Saved in:
Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2018-11, Vol.235 (2), p.182-198
Main Authors: Lanzara, F., Maz’ya, V.
Format: Article
Language:English
Subjects:
DISTRIBUTION
EIGENFUNCTIONS
Eigenvectors
FOURIER TRANSFORMATION
Fourier transforms
INTEGRALS
MATHEMATICAL METHODS AND COMPUTING
Mathematics
Mathematics and Statistics
Operators (mathematics)
SPHERES
SPHERICAL CONFIGURATION
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:A nontrivial example of an eigenfunction in the sense of the theory of distributions for the planar Fourier transform was described by the authors in their previous work. In this paper, a method for obtaining other eigenfunctions is proposed. Positive homogeneous distributions in ℝ n of order − n /2 are considered, and it is shown that F ( ω )| x | − n /2 , | ω | = 1, is an eigenfunction in the sense of the theory of distributions of the Fourier transform if and only if F ( ω ) is an eigenfunction of a certain singular integral operator on the unit sphere of ℝ n . Since Y m , n k ω x − n / 2 , where Y m , n k denote the spherical functions of order m in ℝ n , are eigenfunctions of the Fourier transform, it follows that Y m , n k are eigenfunctions of the above-mentioned singular integral operator. In the planar case, all eigenfunctions of the Fourier transform of the form F ( ω )| x | −1 are described by means of the Fourier coefficients of F ( ω ).
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-018-4067-7