Spectral Functions for the Vector-Valued Fourier Transform
Author
Source
Issue
Vol. 2018, Issue 2018 (31 Dec. 2018), pp.1-17, 17 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2018-10-01
Country of Publication
Egypt
No. of Pages
17
Main Subjects
Abstract EN
A scalar distribution function σ(s) is called a spectral function for the Fourier transform φ^(s)=∫Reitsφ(t)dt (with respect to an interval I⊂R) if for each function φ∈L2(R) with support in I the Parseval identity ∫Rφ^s2dσ(s)=∫Rφt2dt holds.
We show that in the case I=R there exists a unique spectral function σ(s)=(1/2π)s, in which case the above Parseval identity turns into the classical one.
On the contrary, in the case of a finite interval I=(0,b), there exist infinitely many spectral functions (with respect to I).
We introduce also the concept of the matrix-valued spectral function σ(s) (with respect to a system of intervals {I1,I2,…,In}) for the vector-valued Fourier transform of a vector-function φ(t)={φ1(t),φ2(t),…,φn(t)}∈L2(I,Cn), such that support of φj lies in Ij.
The main result is a parametrization of all matrix (in particular scalar) spectral functions σ(s) for various systems of intervals {I1,I2,…,In}.
American Psychological Association (APA)
Mogilevskii, Vadim. 2018. Spectral Functions for the Vector-Valued Fourier Transform. Journal of Function Spaces،Vol. 2018, no. 2018, pp.1-17.
https://search.emarefa.net/detail/BIM-1186748
Modern Language Association (MLA)
Mogilevskii, Vadim. Spectral Functions for the Vector-Valued Fourier Transform. Journal of Function Spaces No. 2018 (2018), pp.1-17.
https://search.emarefa.net/detail/BIM-1186748
American Medical Association (AMA)
Mogilevskii, Vadim. Spectral Functions for the Vector-Valued Fourier Transform. Journal of Function Spaces. 2018. Vol. 2018, no. 2018, pp.1-17.
https://search.emarefa.net/detail/BIM-1186748
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1186748