Decomposition systems with rapidly decaying elements (needlets) based on Hermite functions are introduced and explored. It is proved that the Triebel-Lizorkin and Besov spaces on R induced by Hermite expansions can be characterized in terms of the needlet coefficients. It is also shown that the Hermite Triebel-Lizorkin and Besov spaces are, in general, different from the respective classical sp...

We shall establish two-side explicit inequalities, which are asymptotically sharp up to a constant factor, on the maximum value of |Hk(x)|e −x2/2, on the real axis, where Hk are the Hermite polynomials.

Abstract. The main aim of this paper is to define a new polynomial, say, pseudo hyperbolic matrix functions, pseudo Hermite matrix polynomials and to study their properties. Some formulas related to an explicit representation, matrix recurrence relations are deduced, differential equations satisfied by them is presented, and the important role played in such a context by pseudo Hermite matrix p...

Orthogonal polynomials have been used to produce sharp estimates in Harmonic Analysis in several instances. The first most notorious and original use was in Beckner’s thesis [1], where he proved the sharp Hausdorff-Young inequality using Hermite polynomial expansions. More recently, Foschi [4] used spherical harmonics and Gegenbauer polynomials in his proof of the sharp Tomas-Stein adjoint Four...

Abstract— The concepts and the related aspects of the monomiality principle are presented in this paper to explore different approaches for some classes of orthogonal polynomials. The associated operational calculus introduced by the monomiality principle allows us to reformulate the theory of Hermite, Laguerre and Legendre polynomials from a unified point of view. They are indeed shown to be p...

We introduce the so-called Clifford–Hermite polynomials in the framework of Dunkl operators, based on the theory of Clifford analysis. Several properties of these polynomials are obtained, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the generalized Laguerre polynomials. A link is established with the generalized Hermite polynomials related ...

We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler–Heine type formulas for...

and Applied Analysis 3 The Hermite polynomials are given by Hn x H 2x n n ∑ l 0 ( n l ) 2xHn−l, 1.11 see 23, 24 , with the usual convention about replacing H by Hn. In the special case, x 0, Hn 0 Hn are called the nth Hermite numbers. From 1.11 , we note that d dx Hn x 2n H 2x n−1 2nHn−1 x , 1.12 see 23, 24 , and Hn x is a solution of Hermite differential equation which is given by y′′ − 2xy′ n...

The focus of the research presented in this paper is on a new generalized family degenerate three-variable Hermite–Appell polynomials defined here using fractional derivative. was motivated by investigations Hermite-based Appell introduced R. Alyosuf. We show that, for certain values, well-known polynomials, and are seen as particular cases family. As results investigation, operational rule exp...

Infinitely many Casoratian identities are derived for the Wilson and Askey-Wilson polynomials in parallel to the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials, which were reported recently by the present authors. These identities form the basis of the equivalence between eigenstate adding and deleting Darboux transformations for solvable (discrete) quantum mechanical sys...