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Distributional {D}unkl transform and {D}unkl convolution operators

Jorge J. Betancor — 2006

Bollettino dell'Unione Matematica Italiana

In this paper, that is divided in two parts, we study the distributional Dunkl transform on R. In the first part we investigate the Dunkl transform and the Dunkl convolution operators on tempered distributions. We prove that the tempered distributions defining Dunkl convolution operators on the Schwartz space ƒ are the elements of 𝒪 c ′ , the space of usual convolution operators on S . In the second part we define the distributional Dunkl transform by employing the kernel method. We introduce Frechet...

On Hankel transform and Hankel convolution of Beurling type distributions having upper bounded support

M. Belhadj;  Jorge J. Betancor — 2004

Czechoslovak Mathematical Journal

In this paper we study Beurling type distributions in the Hankel setting. We consider the space ℰ ( w ) ' of Beurling type distributions on ( 0 , ∞ ) having upper bounded support. The Hankel transform and the Hankel convolution are studied on the space ℰ ( w ) ' . We also establish Paley Wiener type theorems for Hankel transformations of distributions in ℰ ( w ) ' .

Multipliers of Hankel transformable generalized functions

Jorge J. Betancor;  Isabel Marrero — 1992

Commentationes Mathematicae Universitatis Carolinae

Let ℋ μ be the Zemanian space of Hankel transformable functions, and let ℋ μ ' be its dual space. In this paper ℋ μ is shown to be nuclear, hence Schwartz, Montel and reflexive. The space O , also introduced by Zemanian, is completely characterized as the set of multipliers of ℋ μ and of ℋ μ ' . Certain topologies are considered on 𝒪 , and continuity properties of the multiplication operation with respect to those topologies are discussed.

A Parseval equation and a generalized finite Hankel transformation

Jorge J. Betancor;  Manuel T. Flores — 1991

Commentationes Mathematicae Universitatis Carolinae

In this paper, we study the finite Hankel transformation on spaces of generalized functions by developing a new procedure. We consider two Hankel type integral transformations h μ and h μ * connected by the Parseval equation ∑ n = 0 ∞ ( h μ f ) ( n ) ( h μ * ϕ ) ( n ) = ∫ 0 1 f ( x ) ϕ ( x ) d x . A space S μ of functions and a space L μ of complex sequences are introduced. h μ * is an isomorphism from S μ onto L μ when μ ≥ - 1 2 . We propose to define the generalized finite Hankel transform h μ ' f of f ∈ S μ ' by 〈 ( h μ ' f ) , ( ( h μ * ϕ ) ( n ) ) n = 0 ∞ 〉 = 〈 f , ϕ 〉 , for ϕ ∈ S μ .

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