MOMENT PROBLEMS AND ORTHOGONAL FUNCTIONS

[摘要] The Caratheodory coefficient problem for an infinite sequence {c(n)} can be formulated as follows. Find a Caratheodory function F(z) (i.e., an analytic function mapping the unit disk D = {z: Absolute value of z < 1} into the right half-plane P = {omega: Re omega > 0} whose Taylor coefficients at a fixed point in D (for convenience the origin) are the {c(n)}. This problem is equivalent to the trigonometric moment problem: find a measure mu(theta) on [-pi, pi] such that integral-pi/-pi e(-intheta) dmu(theta) = c(n), for n = 0, 1, 2,.... These problems are closely related to the theory of Szego polynomials, i.e., orthogonal polynomials on the unit circle T = {z: Absolute value of z = 1}. Let {alpha(n)} be an arbitrary given sequence of not necessarily distinct points in D, and let {omega(n)} be a given sequence of numbers. The Nevanlinna-Pick problem for this situation is to find a Caratheodory function F(z) with the interpolation property F(alpha(n)) = omega(n) for all n. (When points alpha(n) are repeated, the interpolation property involves the appropriate number of Taylor coefficients.) Also this problem is connected with the problem of finding a measure generating ''moments''. The Nevanlinna-Pick problem is related to a theory of rational functions which are orthogonal on T (with poles at the points 1/alpha(n)BAR). This relationship is analogous to and generalizes the relationship between the Caratheodory coefficient problem (or trigonometric moment problem) and polynomials orthogonal on T. The situation when points {alpha(n)} are given on the boundary T of D is also briefly discussed.

[发布日期] 1993-10-29

[效力级别]  Proceedings Paper

[关键词] CARATHEODORY COEFFICIENT PROBLEM;NEVANLINNA-PICK INTERPOLATION PROBLEM;MOMENT PROBLEMS;ORTHOGONAL POLYNOMIALS AND ORTHOGONAL RATIONAL FUNCTIONS ON THE UNIT CIRCLE