In this article, we establish first a geometric Paley–Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the L p → L norm of Dunkl translations in dimension 1. Finally we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension.

Abstract We investigate some spectral properties of a second order differential-difference operator $$J_{\alpha ,\beta }$$ J α , β on $$L^2((-\pi ,\pi ),d\mu _{\alpha , \beta })$$ L <mm...

The $(k,a)$-generalised Fourier transform is the unitary operator defined using $a$-deformed Dunkl harmonic oscillator.The main aim of this paper to prove $L^p$-$L^q$ boundedness $(k, a)$-generalised multipliers. To show we first establish Paley inequality and Hausdorff-Young-Paley for transform. We also demonstrate applications obtained results study well-posedness nonlinear partial differenti...

A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland (CMS) model that decomposes triangularly in terms of the symmetric monomial superfunctions. Many explicit examples are displayed. Furthermore, various ne...

In this paper, three-dimensional Dunkl oscillator models are studied in a generalized form of superintegrable Euclidean Hamiltonian systems to curved ones. These defined based on Hamiltonians, which depend deformation parameter underlying space and involve reflection operators. The corresponding symmetries obtained by the Jordan–Schwinger representations family Cayley–Klein orthogonal algebras ...

By using the Lp(⋅)−Lp(⋅)−boundedness of a maximal operator defined on homogeneous space, it has been shown that B−B−maximal is bounded. In present paper, we aim to bring different approach boundedness generated by generalized translation under continuity assumption p(⋅)p(⋅). It noteworthy mention our weaker than uniform Hölder continuity.

In this paper, the notion of closure operators of matroids  is generalized to fuzzy setting  which is called $M$-fuzzifying closure operators, and some properties of $M$-fuzzifying closure operators are discussed. The $M$-fuzzifying matroid induced by an $M$-fuzzifying closure operator can induce an $M$-fuzzifying closure operator. Finally, the characterizations of $M$-fuzzifying acyclic matroi...

In this paper, we introduce a notion of generalized states from an EQ-algebra E1 to another EQ-algebra E2, which is a generalization of internal states (or state operators) on an EQ-algebra E. Also we give a type of special generalized state from an EQ-algebra E1 to E1, called generalized internal states (or GI-state). Then we give some examples and basic properties of generalized (internal) st...

Given a real and invertible d×d matrix C, we define for k ∈ Zd a generalized translation operator TCk acting on f ∈ L 2(Rd) by (TCkf)(x) = f(x − Ck), x ∈ R . A generalized shift-invariant system is a system of the type {TCjkφj}j∈J,k∈Zd , where {Cj}j∈J is a countable collection of real invertible d×d matrices, and {φj}j∈J ⊂ L 2(Rd). Generalized shift-invariant systems contain the classical wavel...

We continue a program generalizing classical results from the analysis on symmetric cones to Dunkl setting for root systems of type A A ...