A property of series of holomorphic homogeneous polynomials with Hadamard gaps

Abstract

and BQ denote respectively the class of holomorphic functions on D which satisfy |/'(z)| p (l -|z| J ) dxdy is a finite measure, a Carleson measure and a little Carleson measure on D. In this paper we give a higher-dimensional version of Miao's result.

Characterizations for Certain Analytic Functions by Series Expansions With Hadamard Gaps

“…Let D = {z ∈ C : |z| < 1} be the open unit disk in the complex plane C, let H(D) denote the class of functions analytic in the unit disc D, while dA(z) denotes the Lebesgue area measure on the plane, normalized so that A(D) = 1. Let the Green's function of D be defined as g(z, a) = log 1 |ϕa(z)| , where ϕ a (z) = a−z 1−āz is the Möbius transformation related to the point a ∈ D. For 0 < r < 1, let D(a, r) = {r ∈ D : |ϕ a (z)| < r} be the pseudo-hyperbolic disk with center a ∈ D and radius r. In the past few decades both Taylor and Fourier series expansions for various classes of analytic function spaces where the studies are done by the help of Hadamard gap class (see [1,3,10,16] and others).…”