New Convolutions with Hermite Weight Functions

Hermite functions

We define S to be the set of those φ ∈ C∞(R,C) such that pn(φ) < ∞ for all n ≥ 0. S is a complex vector space and each pn is a norm, and because each pn is a norm, a fortiori {pn : n ≥ 0} is a separating family of seminorms. With the topology induced by this family of seminorms, S is a Fréchet space. As well, D : S → S defined by (Dφ)(x) = φ′(x), x ∈ R 1http://individual.utoronto.ca/jordanbell/...

Schwartz functions, Hermite functions, and the Hermite operator

Convolutions for certain analytic functions

Applying the coefficient inequalities of functions f(z) belonging to the subclasses SD(α, β) and KD(α, β) of certain analytic functions in the open unit disk U, two subclasses SD∗(α, β) and KD∗(α, β) are introduced. In this present paper, some interesting convolution properties of functions f(z) in the classes SD∗(α, β) and KD∗(α, β) are discussed by using Shwarz inequality. 2000 Mathematical S...

Gabor (super)frames with Hermite Functions

We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions Hn. Let h = (H0, H1, . . . , Hn) be the vector of the first n+ 1 Hermite functions. We give a complete characterization of all lattices Λ ⊆ R such that the Gabor system {e2πiλ2th(t − λ1) : λ = (λ1, λ2) ∈ Λ} is a frame for L(R,C). As a corollary we obtain sufficient conditions for a single H...

Hermite Orthogonal Rational Functions

We recount previous development of d-fold doubling of orthogonal polynomial sequences and give new results on rational function coefficients, recurrence formulas, continued fractions, Rodrigues’ type formulas, and differential equations, for the general case and, in particular, for the d-fold Hermite orthogonal rational functions.