Bounds on the Segal-Bargmann transform ofL p functions (original) (raw)

Asymptotic behaviours of a class of integral transforms in complex domains

Proceedings of the Edinburgh Mathematical Society, 1989

Zemanian [17] obtained abelian theorems for the Hankel and K-transforms of functions and then extended his results to the corresponding transforms of distributions in the sense of Schwartz [11]. Jones [6] has discussed at length asymptotic behaviours of transforms generalized in his sense. Following the technique of Zemanian many authors have obtained abelian theorems for more general transforms of functions and distributions in the sense of Schwartz. Mention may be made of the works of Joshi and Saxena [7], Lavoine and Misra [8] and Pathak [10]. However, these authors were confined to the transforms of real variables only.

A Universality Property of Gaussian Analytic Functions

Journal of Theoretical Probability, 2011

We consider random analytic functions defined on the unit disk of the complex plane as power series such that the coefficients are i.i.d., complex valued random variables, with mean zero and unit variance. For the case of complex Gaussian coefficients, Peres and Virág showed that the zero set forms a determinantal point process with the Bergman kernel. We show that for general choices of random coefficients, the zero set is asymptotically given by the same distribution near the boundary of the disk, which expresses a universality property. The proof is elementary and general.

An asymptotic approach to proving sufficiency of Stein characterisations

2021

In extending Stein’s method to new target distributions, the first step is to find a Stein operator that suitably characterises the target distribution. In this paper, we introduce a widely applicable technique for proving sufficiency of these Stein characterisations, which can be applied when the Stein operators are linear differential operators with polynomial coefficients. The approach involves performing an asymptotic analysis to prove that only one characteristic function satisfies a certain differential equation associated to the Stein characterisation. We use this approach to prove that all Stein operators with linear coefficients characterise their target distribution, and verify on a case-by-case basis that all polynomial Stein operators in the literature with coefficients of degree at most two are characterising. For X denoting a standard Gaussian random variable and Hp the p-th Hermite polynomial, we also prove, amongst other examples, that the Stein operators for Hp(X), ...

Coefficient inequality for transforms of certain subclass of analytic functions

Acta et Commentationes Universitatis Tartuensis de Mathematica, 2017

The objective of this paper is to obtain the best possible sharp upper bound for the second Hankel functional associated with the kth root transform [f(zk)]1/k of normalized analytic function f(z) when it belongs to certain subclass of analytic functions, defined on the open unit disc in the complex plane using Toeplitz determinants.

Distributional Transformations, Orthogonal Polynomials, and Stein Characterizations

Journal of Theoretical Probability, 2005

A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function's higher order derivatives. The class includes the size and zero bias transformations, and when specializing to weighting by polynomial functions, relates distributional families closed under independent addition, and in particular the infinitely divisible distributions, to the family of transformations induced by their associated orthogonal polynomial systems. For these families, generalizing a well known property of size biasing, sums of independent variables are transformed by replacing summands chosen according to a multivariate distribution on its index set by independent variables whose distributions are transformed by members of that same family. A variety of the transformations associated with the classical orthogonal polynomial systems have as fixed points the original distribution, or a member of the same family with different parameter.

On an Integral Transform of a Class of Analytic Functions

Abstract and Applied Analysis, 2012

For α, γ ≥ 0 and β < 1, let W β α, γ denote the class of all normalized analytic functions f in the open unit disc E {z : |z| < 1} such that e iφ 1 − α 2γ f z /z α − 2γ f z γzf z − β > 0, z ∈ E for some φ ∈ R. It is known Noshiro 1934 and Warschawski 1935 that functions in W β 1, 0 are close-to-convex and hence univalent for 0 ≤ β < 1. For f ∈ W β α, γ , we consider the integral transform F z V λ f z : 1 0 λ t f tz /t dt, where λ is a nonnegative real-valued integrable function satisfying the condition 1 0 λ t dt 1. The aim of present paper is, for given δ < 1, to find sharp values of β such that i V λ f ∈ W δ 1, 0 whenever f ∈ W β α, γ and ii V λ f ∈ W δ α, γ whenever f ∈ W β α, γ .