Best cubature formulas for some classes of continuous functions of two variables

The exact value of the error of a cubature formula is determined for some classes of continuous functions of two variables defined by strictly monotone moduli of continuity.     

On a generalization of compound Newton-Cotes quadrature formulas

We consider quadrature formulas defined by piecewise polynomial interpolation at equidistant nodes, admitting the nodes of adjacent polynomials to overlap, which generalizes the interpolation scheme of the compound Newton-Cotes quadrature formulas. The error constantse _,n in the estimate

Dirichlet problem for pluriholomorphic functions of two complex variables

In this paper the Dirichlet problem for pluriholomorphic functions of two complex variables is investigated. The key point is the relation between pluriholomorphic functions and pluriharmonic functions. The link is constituted by the Fueter-regular functions of one quaternionic variable. Previous results about the boundary values of pluriharmonic functions and new results on L² traces of regular functions are applied to obtain a characterization of the traces of pluriholomorphic functions.     

A generalization of gamma functions and Kronecker's limit formulas

We obtain a “Kronecker limit formula” for the Epstein zeta function. This is done by introducing a generalized gamma function attached to the Epstein zeta function. The methods involve generalizing ideas of Shintani and Stark. We first show that a generalized gamma function appears as the value at s=0 of the first derivative of the associated Epstein zeta function. Then this is used to yield Kronecker's limit formula and its “s=0”-version.     

On the Malgrange condition for complex polynomials of two variables

 We show that if P  ℂ[X, Y] is nonconstant, the set K _∞ (P) of points where the Malgrange condition fails to hold coincides with the roots of a constructible polynomial in one variable. This strengthens weaker related results in the literature. The proof given here is purely analytical and bypasses the algebraic and topological arguments involved in other approaches. Applications to both the real and complex Jacobian Conjectures and to the reducibility of P-z are discussed. Received: 11 October 2001 / Revised version: 2 April 2002 Mathematics Subject Classification (2000): 12D05, 14D06, 14R15, 32B10     

On a diagonal Padé approximation in two complex variables

Summary A special type diagonal Padé approximation for a class of hermitian power series in two variables is related to a canonical strong-operator topology, finite-rank approximation of cyclic operators. The expected convergence of the process (uniform or in measure) is derived from operator theory facts. Paper partially supported by the National Science Foundation Grant DMS-9800666     

Estimating complex covariance by observing two variables at a time

The estimation of covariance matrices is central in array signal processing systems. This note addresses complex covariance estimation for the situation, where the complex data are available only as independent pairwise sets (observations) corresponding to individual elements of the matrix. The formulation for the empirical estimate and the normal maximum likelihood estimate is developed for the general case of different sample sizes for each observation. The approach allows, for example, the estimate of the p by p covariance matrix of a p-port sensor array from a two-port measurement instrument.     

Global solvability of real analytic complex vector fields in two variables

This paper deals with the global solvability of a complex vector field with real analytic coefficients in two real variables. The vector field is assumed to satisfy the Nirenberg-Treves condition (P) for local solvability. Normal forms for the vector field near the one-dimensional orbits are obtained and a generalization of the Riemann-Hilbert problem is considered.