Abstract
A boundary Nevanlinna–Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers \({\kappa_1,\ldots,\kappa_N}\), quaternions p 1,…,p N all of modulus 1, so that the 2-spheres determined by each point do not intersect and p u ≠ 1 for u = 1,…, N, and quaternions s 1,…, s N , we wish to find a slice hyperholomorphic Schur function s so that
and
Our arguments rely on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.
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Alpay, D., Bolotnikov, V., Colombo, F., Sabadini, I.: Self-mappings of the quaternionic unit ball: multiplier properties, Schwarz–Pick inequality, and Nevanlinna–Pick interpolation problem. Indiana Math. J. Math. (to appear). arXiv:1308.2658
Alpay D., Bruinsma P., Dijksma A., de Snoo H.S.V.: Interpolation problems, extensions of symmetric operators and reproducing kernel spaces II. Integral Equ. Oper. Theory 14, 465–500 (1991)
Alpay D., Bruinsma P., Dijksma A., de Snoo H.S.V.: Interpolation problems, extensions of symmetric operators and reproducing kernel spaces II (missing section 3). Integral Equ. Oper. Theory 15, 378–388 (1992)
Alpay, D., Colombo, F., Lewkowicz, I., Sabadini I.: Realizations of slice hyperholomorphic generalized contractive and positive functions. arXiv:1310.1035
Alpay D., Colombo F., Sabadini I.: Krein–Langer factorization and related topics in the slice hyperholomorphic setting. J. Geom. Anal. 24(2), 843–872 (2014)
Alpay D., Colombo F., Sabadini I.: Schur functions and their realizations in the slice hyperholomorphic setting. Integral Equ. Oper. Theory 72, 253–289 (2012)
Alpay D., Colombo F., Sabadini I.: Pontryagin-de Branges–Rovnyak spaces of slice hyperholomorphic functions. J. Anal. Math. 121, 87–125 (2013)
Alpay D., Dijksma A., Langer H., Wanjala G.: Basic boundary interpolation for generalized Schur functions and factorization of rational J-unitary matrix functions. In: Alpay, D., Gohberg, I. (eds.) Interpolation, Schur Functions and Moment Problems. Operator Theory: Advances and Applications, vol. 165, pp. 1–29. Birkhäuser, Basel (2006)
Alpay D., Dubi C.: Boundary interpolation in the ball. Linear Algebra Appl. 340, 33–54 (2002)
Alpay D., Dym H.: Structured invariant spaces of vector valued functions, hermitian forms, and a generalization of the Iohvidov laws. Linear Algebra Appl. 137/138, 137–181 (1990)
Alpay D., Dym H.: On a new class of structured reproducing kernel Hilbert spaces. J. Funct. Anal. 111, 1–28 (1993)
Aronszajn N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)
Ball J.: Models for noncontractions. J. Math. Anal. Appl. 52, 235–259 (1975)
Burckel R.B.: An introduction to classical complex analysis, vol. 1. Birkhäuser, Basel (1979)
Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative Functional Calculus. Theory and Applications of Slice Hyperholomorphic Functions. Progress in Mathematics, vol. 289. Birkhäuser/Springer Basel AG, Basel (2011)
de Branges L.: Some Hilbert spaces of analytic functions I. Trans. Am. Math. Soc. 106, 445–468 (1963)
Dym, H.: J-Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1989)
Fillmore P.A., Williams J.P.: On operator ranges. Adv. Math. 7, 254–281 (1971)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994). Corrected reprint of the 1991 original
Katsnelson, V., Kheifets, A., Yuditskii, P.: An abstract interpolation problem and the extension theory of isometric operators. In: Dym, H., Fritzsche, B., Katsnelson, V., Kirstein, B. (eds.) Topics in Interpolation Theory. Operator Theory: Advances and Applications, vol. 95, pp. 283–297. Birkhäuser, Basel (1997) [Translated from: Operators in Function Spaces and Problems in Function Theory, pp. 83–96 (Naukova-Dumka, Kiev, 1987. Edited by V.A. Marchenko)]
Rovnyak, J.: Characterization of spaces H(M). Unpublished paper, (1968). http://www.people.virginia.edu/~jlr5m/home.html
Sarason D.: Sub-Hardy Hilbert Spaces in the Unit Disk. University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10. Wiley, New York (1994)
Sarason, D.: Nevanlinna–Pick interpolation with boundary data. Integral Equ. Operator Theory 30(2), 231–250 (1998) [Dedicated to the memory of Mark Grigorievich Krein (1907–1989)]
Schwartz L.: Sous espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (noyaux reproduisants). J. Anal. Math. 13, 115–256 (1964)
Zhang F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)
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D. Alpay thanks the Earl Katz family for endowing the chair which supported his research, and the Binational Science Foundation Grant number 2010117. F. Colombo and I. Sabadini acknowledge the Center for Advanced Studies of the Mathematical Department of the Ben-Gurion University of the Negev for the support and the kind hospitality during the period in which part of this paper has been written. D. P. Kimsey gratefully acknowledges funding from a Kreitman postdoctoral fellowship.
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Abu-Ghanem, K., Alpay, D., Colombo, F. et al. Boundary Interpolation for Slice Hyperholomorphic Schur Functions. Integr. Equ. Oper. Theory 82, 223–248 (2015). https://doi.org/10.1007/s00020-014-2184-3
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DOI: https://doi.org/10.1007/s00020-014-2184-3