Abstract:
The concept of ordered families of interpolation problems in the Stieltjes class is introduced. Ordered families are used for the introduction of the concept of limiting interpolation problem in the same class. The limiting interpolation problem is proved to be soluble. A criterion for the complete indeterminacy of a limiting interpolation problem in the Stieltjes class is obtained. All solutions in the completely indeterminate case are described in terms of linear fractional transformations. General constructions are illustrated by the examples of the Stieltjes moment problem and the Nevanlinna–Pick problem in the Stieltjes class.
\Bibitem{Dyu05}
\by Yu.~M.~Dyukarev
\paper Indeterminacy of interpolation problems in the Stieltjes class
\jour Sb. Math.
\yr 2005
\vol 196
\issue 3
\pages 367--393
\mathnet{http://mi.mathnet.ru/eng/sm1276}
\crossref{https://doi.org/10.1070/SM2005v196n03ABEH000884}
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Linking options:
https://www.mathnet.ru/eng/sm1276
https://doi.org/10.1070/SM2005v196n03ABEH000884
https://www.mathnet.ru/eng/sm/v196/i3/p61
This publication is cited in the following 14 articles:
Yury Dyukarev, “The Nevanlinna Formula for Matrix Nevanlinna-Pick Interpolation”, Linear Algebra and its Applications, 2025
Abdon E. Choque-Rivero, Monika Winklmeier, “Explicit Relation Between Two Resolvent Matrices of the Truncated Hausdorff Matrix Moment Problem”, Complex Anal. Oper. Theory, 17:4 (2023)
A. E. Choque-Rivero, B. E. Medina-Hernandez, “On two resolvent matrices of the truncated Hausdorff matrix moment problem”, MAMM, 2022, no. 95, 4
Dyukarev Yu.M., “Entropy Functionals and Their Extremal Values For Solving the Stieltjes Matrix Moment Problem”, Methods Funct. Anal. Topol., 26:1 (2020), 27–38
Abdon E. Choque-Rivero, “Resolvent matrix of the truncated Nevanlinna–Pick matrix interpolation problem via orthogonal rational functions in the Stieltjes class”, Russian Math. (Iz. VUZ), 63:6 (2019), 58–73
Yu. M. Dyukarev, “The zeros of determinants of matrix-valued polynomials that are orthonormal on a semi-infinite or finite interval”, Sb. Math., 209:12 (2018), 1745–1755
Yu. M. Dyukarev, I. Yu. Serikova, “Step-by-step solving of ordered interpolational problem for Stieltjes functions”, Russian Math. (Iz. VUZ), 61:6 (2017), 13–26
Choque-Rivero A.E., “Relations Between the Orthogonal Matrix Polynomials on [a, B], Dyukarev-Stieltjes Parameters, and Schur Complements”, Spec. Matrices, 5:1 (2017), 303–318
Yu. M. Dyukarev, “The criterion for the complete indeterminacy of limiting interpolation problem of Stieltjes type in terms of the orthonormal matrix functions”, Russian Math. (Iz. VUZ), 59:4 (2015), 1–12
Yu. M. Dyukarev, A. E. Choque Rivero, “Criterion for the Complete Indeterminacy of the Nevanlinna–Pick Matrix Problem”, Math. Notes, 96:5 (2014), 651–665
Maxim Derevyagin, “The Jacobi matrices approach to Nevanlinna–Pick problems”, Journal of Approximation Theory, 163:2 (2011), 117
Yu. M. Dyukarev, “A Generalized Stieltjes Criterion for the Complete Indeterminacy of Interpolation Problems”, Math. Notes, 84:1 (2008), 22–37
Yu. M. Dyukarev, I. Yu. Serikova, “Complete indeterminacy of the Nevanlinna–Pick problem in the class S[a,b]”, Russian Math. (Iz. VUZ), 51:11 (2007), 17–29
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