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• W2071081531 abstract "The Carathéodory coefficient problem for an infinite sequence{cn} can be formulated as follows. Find a Carathéodory function F(z) (i.e., an analytic function mapping the unit disk D = {z: |z| < 1} into the right half-plane P = {w: Re w > 0}) whose Taylor coefficients at a fixed points in D (for convenience the origin) are the {cn). This problem is equivalent to the trigonometric moment problem: find a measure μ (θ) on [ −π,π] such that ∫π−πe−inθdμ(θ)=cn for n = 0, 1, 2, …. These problems are closely related to the theory of Szego&#x030B; polynomials, i.e., orthogonal polynomials on the unit circle T = {z: |z| = 1}. Let {αn} be an arbitrary given sequence of not necessarily distinct points in D, and let {wn} be a given sequence of numbers. The Nevanlinna-Pick problem for this situation is to find a Carathéodory function F(z) with the interpolation property F(αn) = wn for all n. (When points αn 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 1αn). This relationship is analogous to and generalizes the relationship between the Carathéodory coefficient problem (or trigonometric moment problem) and polynomials orthogonal on T). The situation when points {αn} are given on the boundary T or D is also briefly discussed." @default.
• W2071081531 created "2016-06-24" @default.
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• W2071081531 date "1993-10-01" @default.
• W2071081531 modified "2023-10-18" @default.
• W2071081531 title "Moment problems and orthogonal functions" @default.
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• W2071081531 doi "https://doi.org/10.1016/0377-0427(93)90315-3" @default.
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