Function spaces between crystal space and Fourier-transform space

1959 ◽  
Vol 112 (1-6) ◽  
Author(s):  
A . L . PATTEBSON
Keyword(s):  
Fourier Transform ◽  
Function Spaces ◽  
Crystal Space

The linear canonical wavelet transform on some function spaces

2018 ◽  
Vol 16 (01) ◽  
pp. 1850010 ◽  
Author(s):  
Yong Guo ◽  
Bing-Zhao Li
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Continuous Operator ◽  
Function Spaces ◽  
Linear Continuous Operator ◽  
Convolution Theorem ◽  
Linear Canonical Transform ◽  
Equivalent Definition ◽  
New Space ◽  
Definition Of

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space [Formula: see text] for convenience. As a generation of FT, it is necessary to detect the linear canonical transform (LCT) on a new space for obtaining the similar properties like FT on [Formula: see text]. Therefore, a space [Formula: see text] generalized from [Formula: see text] is introduced firstly, and further we prove that LCT is a homeomorphism from [Formula: see text] onto itself. The linear canonical wavelet transform (LCWT) is a newly proposed transform based on the convolution theorem in LCT domain. Moreover, we propose an equivalent definition of LCWT associated with LCT and further study some properties of LCWT on [Formula: see text]. Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of [Formula: see text] and [Formula: see text].

A note on holomorphic functions and the Fourier-Laplace transform

2017 ◽  
Vol 120 (2) ◽  
pp. 225 ◽  
Author(s):  
Marcus Carlsson ◽  
Jens Wittsten
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Half Plane ◽  
Classical Problem ◽  
Holomorphic Functions ◽  
Function Spaces ◽  
Classical Function ◽  
Upper Half Plane ◽  
The Fourier Transform ◽  
Half Axis

We revisit the classical problem of when a given function, which is analytic in the upper half plane $\mathbb{C} _+$, can be written as the Fourier transform of a function or distribution with support on a half axis $(-\infty ,b]$, $b\in \mathbb{R} $. We derive slight improvements of the classical Paley-Wiener-Schwartz Theorem, as well as softer conditions for verifying membership in classical function spaces such as $H^p(\mathbb{C} _+)$.

scholarly journals Smoothness and Function Spaces Generated by Homogeneous Multipliers

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Konstantin Runovski ◽  
Hans-Jürgen Schmeisser
Keyword(s):  
Fourier Transform ◽  
Approximation Theory ◽  
Differential Operators ◽  
Function Spaces ◽  
Explicit Representation ◽  
Periodic Functions ◽  
Homogeneous Functions ◽  
Representation Formulas ◽  
The Fourier Transform ◽  
And Function

Differential operators generated by homogeneous functionsψof an arbitrary real orders>0(ψ-derivatives) and related spaces ofs-smooth periodic functions ofdvariables are introduced and systematically studied. The obtained scale is compared with the scales of Besov and Triebel-Lizorkin spaces. Explicit representation formulas forψ-derivatives are obtained in terms of the Fourier transform of their generators. Some applications to approximation theory are discussed.

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