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Journal of Operator Theory

Volume 86, Issue 2, Fall 2021  pp. 275-298.

$B$-spline interpolation problem in Hilbert $C^*$-modules

Authors:  Rasoul Eskandari (1), Michael Frank (2), Vladimir M. Manuilov (3), Mohammad Sal Moslehian (4)
Author institution:(1) Department of Mathematics, Faculty of Science, Farhangian University, Tehran, Iran
(2) Hochschule fuer Technik, Wirtschaft und Kultur (HTWK) Leipzig, Fakultaet Informatik und Medien, PF 301166, D-04251 Leipzig, Germany
(3) Moscow Center for Fundamental and Applied Mathematics, \textit{and} Department of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia
(4) Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran

Summary: We introduce the $B$-spline interpolation problem corresponding to a $C^*$-valued sesquilinear form on a Hilbert $C^*$-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert $C^*$-module is self-dual. Passing to the setting of Hilbert $W^*$-modules, we present our main result by characterizing when the spline interpolation problem for the extended $C^*$-valued sesquilinear form has a solution. Finally, solutions of the $B$-spline interpolation problem for Hilbert $C^*$-modules over $C^*$-ideals of $W^*$-algebras are extensively discussed.

Keywords: $B$-Spline interpolation problem, Hilbert $C^*$-module, self-duality, orthogonal complement

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