Nevanlinna-Pick interpolation by rational functions with a single pole inside the unit disk

We devise an efficient algorithm that, given points z₁,…,z_k in the open unit disk D and a set of complex numbers {f_i,0,f_i,1,…,f_i,ni−1} assigned to each z_i, produces a rational function f with a single (multiple) pole in D, such that f is bounded on the unit circle by a predetermined positive number, and its Taylor expansion at z_i has f_i,0,f_i,1,…,f_i,ni−1 as its first n_i coefficients.     

Strictly positive definite functions on the unit circle

We study strictly positive definite functions on the unit circle in the Euclidean space of dimension two. We develop several conditions pertaining to the determination of such functions. The major result is obtained by considering the set of real numbers as a vector space over the field of rational numbers and then applying the Kronecker approximation theorem and Weyl's criterion on equidistributions.

     

Angular limits of harmonic functions defined in a unit circle

Necessary and sufficient conditions for existence of angular boundary limits at an arbitrary point of a unit circumference are presented in this paper for harmonic functions defined in a unit circle.     

Orthogonal rational functions and quadrature on the unit circle

In this paper we shall be concerned with the problem of approximating the integralI {f}= _– f(eⁱ) d(), by means of the formulaI _n {f}= _j=1 ⁿ A _j ⁽ⁿ⁾ f(x _j ⁽ⁿ⁾ ) where is some finite positive measure. We want the approximation to be so thatI _n{f}=I {f} forf belonging to certain classes of rational functions with prescribed poles which generalize in a certain sense the space of polynomials. In order to get nodes {x _j ⁽ⁿ⁾ } of modulus 1 and positive weightsA _j ⁽ⁿ⁾ , it will be fundamental to use rational functions orthogonal on the unit circle analogous to Szeg polynomials.The work of the first author is partially supported by a research grant from the Belgian National Fund for Scientific Research.     

Factorization of analytic matrix-valued functions

A new method is proposed for calculating the partial indices and the Wiener-Hopf factorization of analytic matrix-valued functions. It has computational advantages with respect to the method proposed by the author earlier. The new method is used to find the divisors of an analytic matrix-valued function a(t) that generate the zeros ofdet a (t). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 324–336, March, 1999.     

On the representation of a class of contractive matrix-valued functions

By a decomposition of L₊²(Cⁿ) in two orthogonal subspaces we obtain a representation of a matrix-valued function of the class $\text{J}$Π, defined by Arov (Darlington realization of matrix-valued functions, Izv. Akad. Nauk. SSSR, Ser. Mat. Tom.37 (1973), No. 6); (Math. USSR Izvestija7 (1973), No. 6, 1295–1326). Real matrix-valued functions of this class play an important role in methods of synthesis of scattering matrices of linear passive n-ports.     

Estimate for the norm of matrix-valued functions

New precise estimates for a norm of matrix-valued functions are obtained. These estimates improve I. Gel'fand's and G. Shilov's estimate for regular functions of matrices and carleman's estimate for resolvents. Applications to differential equations and to spectrum perturbations are considered.     

An extension of the associated rational functions on the unit circle

A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Carathéodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients.The new class under study includes the associated rational functions as a particular case. As a consequence of the previous general analysis, we obtain explicit representations for the associated rational functions of arbitrary order, as well as for the related Carathéodory function. Such representations are used to find new properties of the associated rational functions.     

Quadrature formulas on the unit circle based on rational functions

Quadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szeg polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made.     

The best polynomial approximation of functions analytic in the unit circle

In the Banach space (p,q,), studied by M. I. Gvaradze, we establish a relation between the best polynomial approximation of an entire transcendental function and such important characteristics as its order of growth and its type.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 838–843, June, 1990.     

On sets of uniqueness for harmonic functions in the unit circle

The resutls of this paper show that the structure of sets mentioned in the title is not trivial. For example, it is shown that there exist countalbe sets of uniqueness for logarithmic potential, i.e., closed countable subsets E of the unit circle $\mathbb{T}$ such that $$f \in C(\mathbb{T}),f|_E = 0,U^f |_E = 0 \Rightarrow f \equiv 0.$$ Here $U^f (z) = \tfrac{1}{\pi }\int\limits_0^{2\pi } {f(e^{i\theta } )\log \tfrac{1}{{\left| {z - e^{i\theta } } \right|}}d\theta } $ . On the other hand, it is shown that every countable porous closed subset of $\mathbb{T}$ is a nonuniqueness set. Bibliography: 9 titles.     

Hilbert's boundary-value problem (with coefficients from the Wiener ring) for matrix-valued functions analytic in the unit disk

Hilbert's boundary-value problem is stated and solved for matrix-valued functions, analytic in the unit disk, under the condition that the coefficients and the free term belong to the Wiener ring (_(n×n)). Left standard factorization of the coefficientU(t) leads to the determination of the number of linearly independent solutions of the homogeneous problem and the number and type of conditions under which the inhomogeneous problem is solvable.Translated from Matematicheskie Zametki, Vol. 10, No. 3, pp. 279–286, September, 1971.I wish to express my gratitude to A. V. Batyrev for help in writing this paper.     

Interpolatory quadrature formulas on the unit circle for Chebyshev weight functions

Summary. In this paper, interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions. Positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands. A comparison is made with Szeg? quadrature formulas. Finally, an application to the interval [-1,1] is also carried out. Received February 29, 2000 / Published online August 17, 2001