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.

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 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

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.

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.

scholarly journals The Fourier transform of thick distributions

2020 ◽  
pp. 1-26
Author(s):  
Ricardo Estrada ◽  
Jasson Vindas ◽  
Yunyun Yang
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Canonical Projection ◽  
Test Functions ◽  
Tempered Distributions ◽  
Delta Functions ◽  
The Fourier Transform ◽  
The One ◽  
One Point Compactification ◽  
Logarithmic Type

We first construct a space [Formula: see text] whose elements are test functions defined in [Formula: see text] the one point compactification of [Formula: see text] that have a thick expansion at infinity of special logarithmic type, and its dual space [Formula: see text] the space of sl-thick distributions. We show that there is a canonical projection of [Formula: see text] onto [Formula: see text] We study several sl-thick distributions and consider operations in [Formula: see text] We define and study the Fourier transform of thick test functions of [Formula: see text] and thick tempered distributions of [Formula: see text] We construct isomorphisms [Formula: see text] [Formula: see text] that extend the Fourier transform of tempered distributions, namely, [Formula: see text] and [Formula: see text] where [Formula: see text] are the canonical projections of [Formula: see text] or [Formula: see text] onto [Formula: see text] We determine the Fourier transform of several finite part regularizations and of general thick delta functions.

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 The Meijer transformation of generalized functions

1987 ◽  
Vol 10 (2) ◽  
pp. 267-286 ◽  
Author(s):  
E. L. Koh ◽  
E. Y. Deeba ◽  
M. A. Ali
Keyword(s):  
Bessel Function ◽  
Function Space ◽  
Kernel Function ◽  
Dual Space ◽  
Generalized Functions ◽  
Modified Bessel Function ◽  
Inversion Theorem ◽  
Distributional Sense

This paper extends the Meijer transformation,Mμ, given by(Mμf)(p)=2pΓ(1+μ)∫0∞f(t)(pt)μ/2Kμ(2pt)dt, wherefbelongs to an appropriate function space,μ ϵ (−1,∞)andKμis the modified Bessel function of third kind of orderμ, to certain generalized functions. A testing space is constructed so as to contain the Kernel,(pt)μ/2Kμ(2pt), of the transformation. Some properties of the kernel, function space and its dual are derived. The generalized Meijer transform,M¯μf, is now defined on the dual space. This transform is shown to be analytic and an inversion theorem, in the distributional sense, is established.

scholarly journals Annihilator, Completeness and Convergence of Wavelet System

2007 ◽  
Vol 188 ◽  
pp. 59-105 ◽  
Author(s):  
Kwok-Pun Ho
Keyword(s):  
Function Space ◽  
Dual Space ◽  
Linear Span ◽  
Dual Frame ◽  
Wavelet System ◽  
One Dimensional ◽  
Trivial Intersection ◽  
The One ◽  
Definition Of ◽  
Dyadic Cubes

AbstractWe show that if is a frame and {ψＱ}Ｑ∈Q ∈ ∩ Mα(ℝn) is its dual frame (for the definition of Mα(ℝn), see Definition 2.1), where Q is the collection of dyadic cubes, then for any f ∈ S′(ℝn), there exists a sequence of polynomials, PL,L′,L″, such that(0.1) in the topology of S′(ℝn), where δ(i) = max(2i, 1). We prove this result by explicitly constructing the polynomials PL,L′,L″. Furthermore, using the above result, we assert that the linear span of the one-dimensional wavelet system is dense in a function space if and only if the dual space of this function space has an trivial intersection with the set of polynomials. This is proved by using the annihilator of the one-dimensional wavelet system.

Sign in / Sign up

Export Citation Format

Share Document