Multiple sampling and interpolation in Bergman spaces

We study multiple sampling and interpolation problems with unbounded multiplicities in the weighted Bergman space, both in the hilbertian case and the uniform case.     

WEIGHTED COMPOSITION OPERATORS BETWEEN BERS-TYPE SPACES AND BERGMAN SPACES

This paper characterizes the boundedness and compactness of weighted composition operators between Bers-type space （or little Bers-type space） and Bergman space. Some estimates for the norm of weighted composition operators between those spaces are obtained.     

Harmonic functions of maximal growth: invertibility and cyclicity in Bergman spaces

In the theory of commutative Banach algebras with unit, an element generates a dense ideal if and only if it is invertible, in which case its Gelfand transform has no zeros, and the ideal it generates is the whole algebra. With varying degrees of success, efforts have been made to extend the validity of this result beyond the context of Banach algebras. For instance, for the Hardy space on the unit disk, it is known that all invertible elements are cyclic (an element is cyclic if its polynomial multiples are dense), but cyclic elements need not be invertible. In this paper, we supply examples of functions in the Bergman and uniform Bergman spaces on the unit disk which are invertible, but not cyclic. This answers in the negative questions raised by Shapiro, Nikolskii, Shields, Korenblum, Brown, and Frankfurt.

     

Local sampling set conditions in weighted shift-invariant signal spaces

How to represent a continuous signal in terms of a discrete sequence is a fundamental problem in sampling theory. Most of the known results concern global sampling in shift-invariant signal spaces. But in fact, the local reconstruction from local samples is one of the most desirable properties for many applications in signal processing, e.g. for implementing real-time reconstruction numerically. However, the local reconstruction problem has not been given much attention. In this article, we find conditions on a finite sampling set X such that at least in principle a continuous signal on a finite interval is uniquely and stably determined by their sampling value on the finite sampling set X in shift-invariant signal spaces.     

The behaviour on the rays of functions from the Bergman and Fock spaces

In this note we study the behaviour of holomorphic functions from the Bergman and Fock spaces on the rays of the unit disc U and the complex plane ℂ. We obtain conditions on the finiteness of weighted L ²-integrals of those functions along rays.      

Removable singularities for weighted Bergman spaces

We develop a theory of removable singularities for the weighted Bergman space , where μ is a Radon measure on ℂ. The set A is weakly removable for , and strongly removable for . The general theory developed is in many ways similar to the theory of removable singularities for Hardy H ^p spaces, BMO and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable. In the case when weak and strong removability are the same for all sets, in particular if μ is absolutely continuous with respect to the Lebesgue measure m, we are able to say more than in the general case. In this case we obtain a Dolzhenko type result saying that a countable union of compact removable singularities is removable. When dμ = wdm and w is a Muckenhoupt A _p weight, 1 < p < ∞, the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent p′ = p/(p − 1) and the dual weight w′ = w ^{1/(1 − p)}.     

Compact differences of weighted composition operators from weighted Bergman spaces to weighted-type spaces

We characterize the compactness of differences of weighted composition operators from the weighted Bergman space , 0 < p < ∞, α > −1, to the weighted-type space of analytic functions on the open unit disk D in terms of inducing symbols and . For the case 1 < p < ∞ we find an asymptotically equivalent expression to the essential norm of these operators.     

Difference of composition operators between standard weighted Bergman spaces

We obtain estimates for the norm and essential norm of the difference of two composition operators between certain Bergman spaces. In particular, a necessary and sufficient condition for boundedness and compactness of the operator is established. Finally, we give a sufficient condition for boundedness and compactness of the difference operator between Hardy spaces.     

Schatten-Herz type positive Toeplitz operators on pluriharmonic Bergman spaces

Recently, Schatten-Herz type Toeplitz operators have been studied on the Bergman spaces and the harmonic Bergman spaces. Motivated these results, we study characterizations of positive Toeplitz operators of Schatten-Herz type in terms of averaging functions and Berezin transforms of symbol functions on the ball of pluriharmonic Bergman spaces.     

Berezin transform and Toeplitz operators on harmonic Bergman spaces

For an operator which is a finite sum of products of finitely many Toeplitz operators on the harmonic Bergman space over the half-space, we study the problem: Does the boundary vanishing property of the Berezin transform imply compactness? This is motivated by the Axler-Zheng theorem for analytic Bergman spaces, but the answer would not be yes in general, because the Berezin transform annihilates the commutator of any pair of Toeplitz operators. Nevertheless we show that the answer is yes for certain subclasses of operators. In order to do so, we first find a sufficient condition on general operators and use it to reduce the problem to whether the Berezin transform is one-to-one on related subclasses.     

Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball

We calculate the operator norm of the weighted composition operator from a weighted Bergman space to a weighted-type space on the unit ball of Cⁿ. We also characterize the compactness of the operator.     

Extended Cesàro operators on Bergman spaces

We define an extended Cesàro operator T_g with holomorphic symbol g in the unit ball B of Cⁿ. For a large class of weights w we characterize those g for which T_g is bounded (or compact) from Bergman space L^p_a,w(B) to L^q_a,w(B), 0<p,q<∞. In addition, we obtain some results about equivalent norms, the norm of point evaluation functionals, and the interpolation sequences on L^p_a,w(B).     

多复变数Bergman空间的支配集

本文利用伪双曲度量球对单位球上的Bergman空间的支配集给出完整刻画.证明方法是将Luecking在单位圆盘上的三个重要引理推广到单位球上,从而刻画单位球上的Bergman空间的支配集.     

BERGMAN TYPE PROJECTIONS ON L^p SPACES

The paper is a survey on the action of Bergman type projections on various Lp on three types of holomorphic function spaces： weighted Bergman spaces, the Bloch spaces. The focus is space, and diagonal Besov spaces.     

Asymptotic behaviour of reproducing kernels of weighted Bergman spaces

Let be a domain in , a nonnegative and a positive function on such that is locally bounded, the space of all holomorphic functions on square-integrable with respect to the measure , where is the -dimensional Lebesgue measure, and the reproducing kernel for . It has been known for a long time that in some special situations (such as on bounded symmetric domains with and the Bergman kernel function) the formula

holds true. [This fact even plays a crucial role in Berezin's theory of quantization on curved phase spaces.] In this paper we discuss the validity of this formula in the general case. The answer turns out to depend on, loosely speaking, how well the function can be approximated by certain pluriharmonic functions lying below it. For instance, () holds if is convex (and, hence, can be approximated from below by linear functions), for any function . Counterexamples are also given to show that in general () may fail drastically, or even be true for some and fail for the remaining ones. Finally, we also consider the question of convergence of for , which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of : for instance, if is not real-analytic at some point, then must have zeroes for all sufficiently large.