The Fourier Transform on the Real Line for Functions in L1

1964 ◽  
pp. 1-18
Author(s):  
George Bachman
Keyword(s):  
Fourier Transform ◽  
Real Line ◽  
The Real ◽  
The Fourier Transform

scholarly journals Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1060
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A. del del Olmo
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Heisenberg Group ◽  
Real Line ◽  
Hermite Functions ◽  
Unitary Representations ◽  
Weyl Groups ◽  
The Real ◽  
Symmetry Properties ◽  
The Fourier Transform

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Weyl–Heisenberg group and some of their extensions.

scholarly journals On the Fourier transforms

1985 ◽  
Vol 31 (2) ◽  
pp. 171-179
Author(s):  
Hwai-chiuan Wang
Keyword(s):  
Fourier Transform ◽  
Convex Function ◽  
Real Line ◽  
Integrable Function ◽  
Fourier Transforms ◽  
The Real ◽  
Factorization Problem ◽  
Several Variables ◽  
The Fourier Transform

In this article we give a new proof of the theorem that a positive even convex function on the real line, which vanishes at infinity, is the Fourier transform of an integrable function. Related results in several variables are also proved. As an application of our results we solve the factorization problem of Sobolev algebras.

Fourier Transforms of Unbounded Measures

1979 ◽  
Vol 31 (6) ◽  
pp. 1281-1292 ◽  
Author(s):  
James Stewart
Keyword(s):  
Fourier Transform ◽  
Real Line ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Stieltjes Transform ◽  
The Real ◽  
Bounded Complex ◽  
The Fourier Transform ◽  
Image Position

1. Introduction. One of the basic objects of study in harmonic analysis is the Fourier transform (or Fourier-Stieltjes transform) μ of a bounded (complex) measure μ on the real line R:(1.1)More generally, if μ is a bounded measure on a locally compact abelian group G, then its Fourier transform is the function(1.2)where Ĝ is the dual group of G and One answer to the question “Which functions can be represented as Fourier transforms of bounded measures?” was given by the following criterion due to Schoenberg [11] for the real line and Eberlein [5] in general: f is a Fourier transform of a bounded measure if and only if there is a constant M such that(1.3)for all ϕ ∈ L1(G) where

TRANSMUTATION OPERATORS AND PALEY–WIENER THEOREM ASSOCIATED WITH A CHEREDNIK TYPE OPERATOR ON THE REAL LINE

2010 ◽  
Vol 08 (04) ◽  
pp. 387-408 ◽  
Author(s):  
MOHAMED ALI MOUROU
Keyword(s):  
Real Line ◽  
Difference Operators ◽  
Generalized Convolution ◽  
One Dimensional ◽  
The Real ◽  
First Order ◽  
Wiener Theorem ◽  
The Fourier Transform ◽  
The One ◽  
Transmutation Operators

We consider a singular differential-difference operator Λ on the real line which generalizes the one-dimensional Cherednik operator. We construct transmutation operators between Λ and first-order regular differential-difference operators on ℝ. We exploit these transmutation operators, firstly to establish a Paley–Wiener theorem for the Fourier transform associated with Λ, and secondly to introduce a generalized convolution on ℝ tied to Λ.

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