Real Separated Algebraic Curves, Quadrature Domains, Ahlfors Type Functions and Operator Theory

The aim of this paper is to inter-relate several algebraic and analytic objects, such as real-type algebraic curves, quadrature domains, functions on them and rational matrix functions with special properties, and some objects from Operator Theory, such as vector Toeplitz operators and subnormal operators. Our tools come from operator theory, but some of our results have purely algebraic formulation. We make use of Xia's theory of subnormal operators and of the previous results by the author in this direction. We also correct (in Section 5) some inaccuracies in two papers by the author in Revista Matematica Iberoamericana (1998).Comment: 43 pages, 2 figures; zip archiv

An Overdetermined Problem in Potential Theory

We investigate a problem posed by L. Hauswirth, F. H\'elein, and F. Pacard, namely, to characterize all the domains in the plane that admit a "roof function", i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data. Under some a priori assumptions, we show that the only three examples are the exterior of a disk, a halfplane, and a nontrivial example. We show that in four dimensions the nontrivial simply connected example does not have any axially symmetric analog containing its own axis of symmetry.Comment: updated version. 20 pages, 3 figure

On Weak Tractability of the Clenshaw-Curtis Smolyak Algorithm

We consider the problem of integration of d-variate analytic functions defined on the unit cube with directional derivatives of all orders bounded by 1. We prove that the Clenshaw Curtis Smolyak algorithm leads to weak tractability of the problem. This seems to be the first positive tractability result for the Smolyak algorithm for a normalized and unweighted problem. The space of integrands is not a tensor product space and therefore we have to develop a different proof technique. We use the polynomial exactness of the algorithm as well as an explicit bound on the operator norm of the algorithm.Comment: 18 page

Generalized intelligent states of the su(N) algebra

Schr\" odinger-Robertson uncertainty relation is minimized for the quadrature components of Weyl generators of the algebra $su(N)$. This is done by determining explicit Fock-Bargamann representation of the $su(N)$ coherent states and the differential realizations of the elements of $su(N)$. New classes of coherent and squeezed states are explicitly derived

Complexity in complex analysis

We show that the classical kernel and domain functions associated to an n-connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable. We characterize those domains for which the classical functions are given by rational combinations of only two or fewer functions of one complex variable. Such domains turn out to have the property that their classical domain functions all extend to be meromorphic functions on a compact Riemann surface, and this condition will be shown to be equivalent to the condition that an Ahlfors map and its derivative are algebraically dependent. We also show how many of these results can be generalized to finite Riemann surfaces.Comment: 30 pages, to appear in Advances in Mat

Tensor product approximations of high dimensional potentials

The paper is devoted to the efficient computation of high-order cubature formulas for volume potentials obtained within the framework of approximate approximations. We combine this approach with modern methods of structured tensor product approximations. Instead of performing high-dimensional discrete convolutions the cubature of the potentials can be reduced to a certain number of one-dimensional convolutions leading to a considerable reduction of computing resources. We propose one-dimensional integral representions of high-order cubature formulas for n-dimensional harmonic and Yukawa potentials, which allow low rank tensor product approximations.Comment: 20 page

Polyharmonic Hardy Spaces on the Klein-Dirac Quadric with Application to Polyharmonic Interpolation and Cubature Formulas

In the present paper we introduce a new concept of Hardy type space naturally defined on the Klein-Dirac quadric. We study different properties of the functions belonging to these spaces, in particular boundary value problems. We apply these new spaces to polyharmonic interpolation and to interpolatory cubature formulas.Comment: 32 page