Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[A] N. Aronszajn,Theory of reproducing kernels, Trans. Amer. Math. Soc.68 (1950), 337–404.
[B] J. A. Ball,Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions, Integral Equations Operator Theory6 (1983), 804–840.
[BGR] J. A. Ball, I. C. Gohberg and L. Rodman,Boundary Nevanlinna-Pick interpolation for rational matrix functions, J. Math. Estim. Control1 (1991), 131–164.
[BH] J. A. Ball and J. W. Helton,Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: parametrization of the set of all solutions, Integral Equations Operator Theory9 (1986), 155–203.
[Bo] V. Bolotnikov,Tangential Carathèodory-Fejér interpolation for Stieltjes functions at real points, Z. Anal. Anwendungen13 (1994), 111–136.
[CF1] R. E. Curto and L. A. Fialkow,Recursiveness, positivity, and truncated moment problems, Houston J. Math.17 (1991), 603–635.
[CF2] R. E. Curto and L. A. Fialkow,Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc.,119, no 568 (1996), x + 52 pp.
[CF3] R. E. Curto and L. A. Fialkow,Flat extensions of positive moment matrices: relations in analytic or conjugate terms, Nonselfadjoint operator algebras, operator theory, and related topics, Oper. Theory Adv. Appl.104 (1998), 59–82.
[CF4] R. E. Curto and L. A. Fialkow,The truncated complex K-moment problem, Trans. Amer. Math. Soc.352 (2000), 2825–2855.
[CF5] R. E. Curto and L. A. Fialkow,The quadratic moment problem for the unit disk and unit circle, Integral Equations Operator Theory (to appear).
[CF6] R. E. Curto and L. A. Fialkow,Flat extensions of positive moment matrices: recursively generated relations, Mem. Amer. Math. Soc.136, no. 648 (1998), x + 56 pp.
[D] H. Dym,On Hermitian block Hankel matrices, matrix polynomials, the Hamburger moment problem, interpolation and maximum entropy, Integral Equations Operator Theory12 (1989), 757–812.
[dB] L. de Branges,Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood Cliffs, N. J., 1968.
[F] I. P. Fedchina,A criterion for the solvability of the Nevanlinna-Pick tangent problem, Mat. Issled.7, no 4(26) (1972), 213–227.
[G1] D. R. Georgijević,Interpolation problems of Loewner type with finitely many nodes, Acta Sci. Math. (Szeged)64 (1998), 81–96.
[G2] D. R. Georgijević,Radial limits in some analytic spaces, Mathematica Montisnigri9 (1998), 65–80.
[G3] D. R. Georgijević,Solvability condition for a boundary value interpolation problem of Loewner type, J. Anal. Math.74 (1998), 213–234.
[H] G. Hamburger,Über eine Erweiterung des Stieltjes'schen Momentenproblems, Math. Ann.81 (1920), 235–319;82 (1921), 120–164; 168–187.
[Ho] K. Hoffman,Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N. J., 1962.
[K1] I. V. Kovalishina,Analytic theory of a class of interpolation problems, Math. USSR Izvestya22 (1984), 419–463.
[K2] I. V. Kovalishina,The multiple interpolational boundary value problem for compressing matrix-functions in the unit circle, Teor. Funktsii Funktsional. Anal. I Prilozhen.51 (1989), 38–55.
[L] K. Löwner,Über monotone Matrixfunktionen, Math. Z.38 (1934), 177–216.
[N] R. Nevanlinna,Über beschränkte analytische Funktionen, Ann. Acad. Sci. Fenn. Ser. A32, no 7 (1929), 1–75.
[RR] M. Rosenblum and J. Rovnyak,An operator theoretic approach to theorems of the Pick-Nevanlinna and Loewner types II, Integral Equations Operator Theory5 (1982), 870–887.
[S] D. Sarason,Nevanlinna-Pick interpolation with boundary data, Integral Equations Operator Theory30 (1998), no 2, 231–250.
[SzNK] B. Sz.-Nagy and A. Korànyi,Relations d'un problème de Nevanlinna et Pick avec la théorie des opérateurs de l'espace Hilbertien, Acta Math. Acad. Sci. Hungar.7 (1956), 295–302.