The n-Dimensional Distributional Hankel Transformation

1975 ◽  
Vol 27 (2) ◽  
pp. 423-433 ◽  
Author(s):  
E. L. Koh
Keyword(s):  
Function Space ◽  
Generalized Functions ◽  
One Dimension ◽  
Dimensional Case ◽  
Hankel Transformation ◽  
Test Function ◽  
Test Function Space

The Hankel transformation was extended to certain generalized functions of one dimension [1; 2; 3]. In this paper, we develop the n-dimensional case corresponding to [1]. The procedure in [1] is briefly as follows:A test function space Hμ is constructed on which the μth order Hankel transformation hμ defined byis an automorphism whenever μ ≧ —1/2. The generalized transformation hμ' is then defined on the dual Hμ' as the adjoint of hμ through a Parseval relation, i.e.

Continuous wavelet transform involving canonical convolution

2019 ◽  
Vol 13 (06) ◽  
pp. 2050104
Author(s):  
Zamir Ahmad Ansari
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Convolution Operator ◽  
Linear Canonical Transform ◽  
Continuous Wavelet ◽  
Test Function ◽  
Test Function Space

The main objective of this paper is to study the continuous wavelet transform in terms of canonical convolution and its adjoint. A relation between the canonical convolution operator and inverse linear canonical transform is established. The continuity of continuous wavelet transform on test function space is discussed.

scholarly journals Periodic Tempered Distributions of Beurling Type and Periodic Ultradifferentiable Functions with Arbitrary Support

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Byung Keun Sohn
Keyword(s):  
Function Space ◽  
Dual Space ◽  
Ultradifferentiable Functions ◽  
Tempered Distributions ◽  
Test Function ◽  
Test Function Space

Let Sω′(R) be the space of tempered distributions of Beurling type with test function space Sω(R) and let Eω,p be the space of ultradifferentiable functions with arbitrary support having a period p. We show that Eω,p is generated by Sω(R). Also, we show that the mapping Sω(R)→Eω,p is linear, onto, and continuous and the mapping Sω,p′(R)→Eω,p′ is linear and onto where Sω,p′(R) is the subspace of Sω′(R) having a period p and Eω,p′ is the dual space of Eω,p.

scholarly journals Test function space for Wick power series

2000 ◽  
Vol 123 (3) ◽  
pp. 709-725 ◽  
Author(s):  
A. G. Smirnov ◽  
M. A. Solov'ev
Keyword(s):  
Power Series ◽  
Function Space ◽  
Test Function ◽  
Test Function Space

scholarly journals Paley–Wiener properties for spaces of entire functions

2020 ◽  
Vol 10 (2) ◽  
Author(s):  
Elmira Nabizadeh ◽  
Christine Pfeuffer ◽  
Joachim Toft
Keyword(s):  
Function Space ◽  
Entire Functions ◽  
Test Function ◽  
Test Function Space

Abstract We deduce Paley–Wiener results in the Bargmann setting. At the same time we deduce characterisations of Pilipović spaces of low orders. In particular we improve the characterisation of the Gröchenig test function space $$\mathcal {H}_{\flat _1}=\mathcal {S}_C$$ H ♭ 1 = S C , deduced in Toft (J Pseudo-Differ Oper Appl 8:83–139, 2017).

On the Bogolyubov endomorphisms of the renormalized square of white noise algebra

2021 ◽  
Author(s):  
Habib Rebei ◽  
Slaheddine Wannes
Keyword(s):  
White Noise ◽  
Function Space ◽  
The Other ◽  
Linear Transformations ◽  
Test Function ◽  
Canonical Commutation Relations ◽  
Arbitrary Complex ◽  
Borel Functions ◽  
Test Function Space ◽  
Complex Valued

We introduce the quadratic analogue of the Bogolyubov endomorphisms of the canonical commutation relations (CCR) associated with the re-normalized square of white noise algebra (RSWN-algebra). We focus on the structure of a subclass of these endomorphisms: each of them is uniquely determined by a quadruple [Formula: see text], where [Formula: see text] are linear transformations from a test-function space [Formula: see text] into itself, while [Formula: see text] is anti-linear on [Formula: see text] and [Formula: see text] is real. Precisely, we prove that [Formula: see text] and [Formula: see text] are uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text], into itself. Under some additional analytic conditions on [Formula: see text] and [Formula: see text], we discover that we have only two equivalent classes of Bogolyubov endomorphisms, one of them corresponds to the case [Formula: see text] and the other corresponds to the case [Formula: see text]. Finally, we close the paper by building some examples in one and multi-dimensional cases.

The theory of weak functions. I

1963 ◽  
Vol 276 (1365) ◽  
pp. 149-167 ◽  
Keyword(s):  
Continuous Function ◽  
Weak Convergence ◽  
Fourier Transforms ◽  
Generalized Functions ◽  
One Dimension ◽  
Test Functions ◽  
Weak Derivative

This paper develops the theory of distributions or generalized functions without any reference to test functions and with no appeal to topology, apart from the concept of weak convergence. In the calculus of weak functions, which is so obtained, a weak function is always a weak derivative of a numerical continuous function, and the fundamental techniques of multiplication, division and passage to a limit are considerably simplified. The theory is illustrated by application to Fourier transforms. The present paper is restricted to weak functions in one dimension. The extension to several dimensions will be published later.

scholarly journals The second dual of C0 (S, A)

1994 ◽  
Vol 56 (3) ◽  
pp. 303-313
Author(s):  
Stephen T. L. Choy ◽  
James C. S. Wong
Keyword(s):  
Function Space ◽  
Simple Proof ◽  
Representation Theorem ◽  
Generalized Functions ◽  
Arens Product ◽  
Nikodym Property ◽  
Second Dual ◽  
Vector Valued Function ◽  
Vector Valued

AbstractThe second dual of the vector-valued function space C0(S, A) is characterized in terms of generalized functions in the case where A* and A** have the Radon-Nikodým property. As an application we present a simple proof that C0 (S, A) is Arens regular if and only if A is Arens regular in this case. A representation theorem of the measure μh is given, where , h ∈ L∞ (|μ;|, A**) and μh is defined by the Arens product.

Representation of Symmetries and Observables in Functional Hilbert Space. II.

1972 ◽  
Vol 27 (1) ◽  
pp. 7-22 ◽  
Author(s):  
A. Rieckers
Keyword(s):  
Hilbert Space ◽  
Relativistic Quantum ◽  
Preceding Paper ◽  
Complex Hilbert Space ◽  
Unbounded Operators ◽  
Test Function ◽  
Test Function Space ◽  
Quantum Observables ◽  
Set Up ◽  
Real Test

Abstract The representation of infinitesimal generators corresponding to the group representation dis-cussed in the preceding paper is analyzed in the Hilbert space of functionals over real test functions. Explicit expressions for these unbounded operators are constructed by means of the functio-nal derivative and by canonical operator pairs on dense domains. The behaviour under certain basis transformations is investigated, also for non-Hermitian generators. For the Hermitian ones a common, dense domain is set up where they are essentially selfadjoint. After having established a one-to-one correspondence between the real test function space and a complex Hilbert space the theory of quantum observables is applied to the functional version of a relativistic quantum field theory.

scholarly journals Representation of functions as the Post-Widder inversion operator of generalized functions

1984 ◽  
Vol 7 (2) ◽  
pp. 371-396 ◽  
Author(s):  
R. P. Manandhar ◽  
L. Debnath
Keyword(s):  
Representation Theory ◽  
Function Space ◽  
Fundamental Theorem ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Representation Theorems ◽  
The Real ◽  
Inversion Operator ◽  
Inversion Theory ◽  
Necessary And Sufficient

A study is made of the Post-Widder inversion operator to a class of generalized functions in the sense of distributional convergence. Necessary and sufficient conditions are proved for a given function to have the representation as therth operate of the Post-Widder inversion operator of generalized functions. Some representation theorems are also proved. Certain results concerning the testing function space and its dual are established. A fundamental theorem regarding the existence of the real inversion operator (1.6) withr=0is proved in section4. A classical inversion theory for the Post-Widder inversion operator with a few other theorems which are fundamental to the representation theory is also developed in this paper.

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