Nevanlinna-Pick interpolation with boundary data

Versions of the Nevanlinna-Pick interpolation problem with boundary interpolation nodes and boundary interpolated values are investigated.     

On the Nevanlinna-Pick interpolation problem for generalized stieltjes functions

The solutions of the Nevanlinna-Pick interpolation problem for generalized Stieltjes matrix functions are parametrized via a fractional linear transformation over a subset of the class of classical Stieltjes functions. The fractional linear transformation of some of these functions may have a pole in one or more of the interpolation points, hence not all Stieltjes functions can serve as a parameter. The set of excluded parameters is characterized in terms of the two related Pick matrices.Dedicated to the memory of M. G. Kreîn     

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.     

A fast algorithm for scalar Nevanlinna-Pick interpolation

Summary In this paper, we derive a fast algorithm for the scalar Nevanlinna-Pick interpolation. Givenn distinct pointsz _i in the unit disk |z|<1 andn complex numbersw _i satisfying the Pick condition for 1in, the new Nevanlinna-Pick interpolation algorithm requires onlyO(n) arithmetic operations to evaluate the interpolatory rational function at a particular value ofz, in contrast to the classical algorithm which requiresO(n ²) arithmetic operations to compute the so-called Fenyves array (which is inherent in the classical algorithm). The new algorithm bypasses the generation of the Fenyves array to speed up the computation, and also yields a parallel scheme requiring onlyO(logn) arithmetic operations on a concurrent-read, exclusive-write parallel random access machine withn processors. We must remark that the rational functionf(z) computed by the new algorithm is one degree higher than the function computed by the classical algorithm.Supported in part by the US Army Research Office Grant No. DAAL03-91-G-0106     

The two-point spectral Nevanlinna-Pick problem

We present an analytic solution of certain special cases of the spectral Nevanlinna-Pick problem for two interpolation points and 2×2 matrices.     

Tangential Nevanlinna-Pick Interpolation and Its Connection with Hamburger Matrix Moment Problem

This paper is concerned with the solution of a certain tangential Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called block Hankel vector method to establish two intrinsic connections between the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the truncated Hamburger matrix moment problem associated with the block Hankel vector under consideration: one is a congruent relationship between their information matrices, and the other is a divisor-remainder connection between their solutions. These investigations generalize our previous work on the Nevanlinna-Pick interpolation and power matrix moment problem.     

Tangential interpolation with symmetries and two-point interpolation problem for matrix-valued H2-functions

We describe all solutions of the two-sided tangential interpolation problem in the class of matrix-valued Hardy functions when symmetries are added: these symmetries are defined in terms of involutions ofH ₂. The obtained results are applied to a one-sided two-points tangential interpolation for matrix functions.The research of this author is partially supported by the NSF Grant DMS 9500924 and by the Binational United States-Israel Foundation Grant 9400271.     

Mixed Löwner and Nevanlinna-Pick Interpolation

A mixed type, L?wner and Nevanlinna-Pick directional two-sided interpolation problem is considered. A necessary and sufficient condition for the problem to have a solution is established, in terms of properties of the Pick kernel to the problem. As well, a parametrization of the set of all real rational solutions of minimal degree is given. The corresponding Nevanlinna-Pick boundary-interior interpolation problem is also considered and a solvability condition for it is obtained. The approach to the problem is via functional Hilbert spaces.     

An interpolation problem in the class of stieltjes functions and its connection with other problems

This paper contains the proofs and some development of results that were published without proofs in [KN2]. It is completed with comments added by the second author explaining how these results became the basis of statements and the solutions of problems in the theory of entire functions, in the moment problem, in direct and inverse problems of the spectral theory of nonhomogeneous strings and in other problems.     

The Webster scalar curvature problem on the three dimensional CR manifolds

In this paper, we prove some existence results for the Webster scalar curvature problem on the three dimensional CR compact manifolds locally conformally CR equivalent to the unit sphere S³ of C². Our methods are based on the techniques related to the theory of critical points at infinity.     

Classes of pairs of meromorphic matrix valued functions generated by nonnegative kernels and associated Nevanlinna-Pick problems

Families of pairs of matrix-valued meromorphic functions (,P) depending on two parameters andP are introduced. They are the projective analogues of classes of functions studied in [1] and include as special cases the projective Schur, Nevanlinna and Carathéodory classes. A two sided Nevanlinna-Pick interpolation problem is defined and solved in (,P), using the fundamental matrix inequality method.     

Discrete time-variant interpolation as classical interpolation with an operator argument

Necessary and sufficient conditions are derived for the existence of solutions to discrete time-variant interpolation problems of Nevanlinna-Pick and Nudelman type. The proofs are based on a reduction scheme which allows one to treat these time-variant interpolation problems as classical interpolation problems for operator-valued functions with operator arguments. The latter ones are solved by using the commutant lifting theorem.     

Romberg integration as a problem in interpolation theory

A complete derivation of Romberg integration for an arbitrary sequence of integration steplenghts, using classical interpolation theory only, is given. An explicit expression for the error is derived using Lagrange interpolation. From the general theory developed, several previous known results may be derived as special cases.     

On an operator approach to interpolation problems for Stieltjes functions

A general interpolation problem for operator-valued Stieltjes functions is studied using V. P. Potapov's method of fundamental matrix inequalities and the method of operator identities. The solvability criterion is established and under certain restrictions the set of all solutions is parametrized in terms of a linear fractional transformation. As applications of a general theory, a number of classical and new interpolation problems are considered.     

Nudelman interpolation and the band method

In previous work the authors developed a new addition of the band method based on a Grassmannian approach for solving a completion/extension problem in a general, abstract framework. This addition allows one to obtain a linear fractional representation of all solutions of the abstract completion problem from special extensions which are not necessarily band extensions (for the positive case) or triangular extensions (for the contractive case). In this work we extend this framework to a somewhat more general setting and show how one can obtain formulas for the required special extensions from solutions of a system of linear equations. As an application we show how the formalism can be applied to the bitangential Nevanlinna-Pick interpolation problem, a case which, up to now, was not amenable to the band method.The first author was partially supported by National Science Foundation grant DMS-9500912.     

Geometric conditions for interpolation in weighted spaces of entire functions

We use L² estimates for the equation to find geometric conditions on discrete interpolating varieties for weighted spaces A_p(ℂ) of entire functions such that |f(z)|≤Ae^Bp(z) for some A, B>0. In particular, we give a characterization when p(z)=e^|z| and more generally, when In p(e^r) is convex andIn p(r) is concave. Acknowledgements and Notes. The author wishes to thank X. Massaneda for useful talks and remarks.     

Matrix valued interpolation and truncated Hamburger moment problems

A solvability condition for matrix valued directional single-node interpolation problems of Loewner type is established, in terms of properties of Pick kernel. As a consequence, a solvability condition for matrix valued directional truncated Hamburger moment problems is obtained.