The standard Berezin and Berezin-Toeplitz quantizations on a Kähler manifold are based on operator symbols and on Toeplitz operators, respectively, on weighted L-spaces of holomorphic functions (weighted Bergman spaces). In both cases, the construction basically uses only the fact that these spaces have a reproducing kernel. We explore the possibilities of using other function spaces with reproducing kernels instead, such as L-spaces of harmonic functions, Sobolev spaces, Sobolev spaces of holomorphic...
We exhibit a class of bounded, strongly convex Hartogs domains with real-analytic boundary which are not Lu Qi-Keng, i.e. whose Bergman kernel function has a zero.
The basic idea of this paper is to use the covariance of a partial differential operator under a suitable group action to determine suitable associated Green’s functions. For instance, we offer a new proof of a formula for Green’s function of the mth power of the ordinary Laplace’s operator Δ in the unit disk found in a recent paper (Hayman-Korenblum, J. Anal. Math. 60 (1993), 113-133). We also study Green’s functions associated with mth powers of the Poincaré invariant Laplace operator . It turns...
It is well known that one can often construct a star-product by expanding the product of two Toeplitz operators asymptotically into a series of other Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin-Toeplitz quantization. We show that one can obtain in a similar way in fact any star-product which is equivalent to the Berezin-Toeplitz star-product, by using instead of Toeplitz operators other suitable mappings from compactly supported smooth functions...
We give a complete description of the boundary behaviour of the generalized
hypergeometric functions, introduced by Faraut and Koranyi, on Cartan domains of rank 2.
The main tool is a new integral representation for some spherical polynomials, which may
be of independent interest.
Let be a measure on a domain in such that the Bergman space
of holomorphic functions in possesses a reproducing kernel and
. The Berezin transform associated to is the
integral operator
The number can be interpreted as a certain mean value of around , and
functions satisfying as functions having a certain mean-value property. In this
paper we investigate the boundary behavior of , the existence of functions satisfying and having...
We consider Bergman projections and some new generalizations of them on weighted -spaces. A new reproducing formula is obtained. We show the boundedness of these projections for a large family of weights v which tend to 0 at the boundary with a polynomial speed. These weights may even be nonradial. For logarithmically decreasing weights bounded projections do not exist. In this case we instead consider the projective description problem for holomorphic inductive limits.
In an earlier paper, the first two authors have shown that the convolution of a function continuous on the closure of a Cartan domain and a -invariant finite measure on that domain is again continuous on the closure, and, moreover, its restriction to any boundary face depends only on the restriction of to and is equal to the convolution, in , of the latter restriction with some measure on uniquely determined by . In this article, we give an explicit formula for in terms of ,...
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