Bergman-type Spaces Research Articles

On the structure of Bergman and poly-Bergman spaces

Let II be the upper half-plane inℂ, consider the Bergman space $$\mathcal{A}^2 (II)$$ , the subspace of all analytic functions fromL 2(II). The complete decomposition ofL 2(II) onto Bergman and Bergman type spaces of poly-analytic and poly-anti-analytic functions is obtained. The orthogonal Bergman type projections onto each of these subspaces are described. Connections with the Hardy spaces and the Szego projections are established.

Traces of certain classes of holomorphic functions on finite unions of Carleson sequences

We give a method allowing the generalization of the description of trace spaces of certain classes of holomorphic functions on Carleson sequences to finite unions of Carleson sequences. We apply the result to different classes of spaces of holomorphic functions such as Hardy classes and Bergman type spaces.

On Unconditional Polynomial Bases in L p and Bergman Spaces

In this paper we consider unconditional bases inL p(T), 1<p<∞,p ≠ 2, consisting of trigonometric polynomials. We give a lower bound for the degree of polynomials in such a basis (Theorem 3.4) and show that this estimate is best possible. This is applied to the Littlewood-Paley-type decompositions. We show that such a decomposition has to contain exponential gaps. We also consider unconditional polynomial bases inH p as bases in Bergman-type spaces and show that they provide explicit isomorphisms between Bergman-type spaces and natural sequences spaces.

On unconditional polynomial bases inL p and bergman spaces

In this paper we consider unconditional bases inL p(T), 1<p<∞,p ≠ 2, consisting of trigonometric polynomials. We give a lower bound for the degree of polynomials in such a basis (Theorem 3.4) and show that this estimate is best possible. This is applied to the Littlewood-Paley-type decompositions. We show that such a decomposition has to contain exponential gaps. We also consider unconditional polynomial bases inH p as bases in Bergman-type spaces and show that they provide explicit isomorphisms between Bergman-type spaces and natural sequences spaces.

Cyclicity in Bergman-type spaces

Cyclicity in Bergman-type spaces

Function spaces and reproducing kernels on bounded symmetric domains

There are several natural Hilbert spaces of holomorphic functions on a bounded symmetric domain. Such are the Bergman-type spaces, on which the holomorphic discrete series operates, and the Hardy-type spaces which are related to the analytic continuation of the holomorphic discrete series. Also closely related is the Bergmann space of entire functions on the ambient @” which arises as the closure of the polynomials with respect to a natural inner product. The space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group K of the domain. The main facts about this decomposition were proved by Schmid [21]; for another proof see [22]. Each irreducible subspace contains a unique normalized L-invariant (“spherical”) polynomial, where L is the isotropy group of the Shilov boundary in K. Our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the Hilbert spaces considered. For the domains of classical type they were considered by Hua [9] and for some of the Hilbert spaces this was done before by Upmeier [24] using different methods; for certain others there are partial results in [22]. We are able to do this in a fairly simple unified way by making strong use of Gindikin’s generalized Gamma function [S]. Next we obtain a description of the reproducing kernels of the K-irreducible subspaces in each of our Hilbert spaces, and an expansion in terms of these for