On the tangential interpolation problem for 𝐻₂ functions

Alpay, Daniel; Bolotnikov, Vladimir; Peretz, Yossi

doi:10.1090/S0002-9947-1995-1277087-2

On the tangential interpolation problem for $H_ 2$ functions
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by Daniel Alpay, Vladimir Bolotnikov and Yossi Peretz

Trans. Amer. Math. Soc. 347 (1995), 675-686

DOI: https://doi.org/10.1090/S0002-9947-1995-1277087-2

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Abstract:

The aim of this paper is to solve a matrix-valued version of the Nevanlinna-Pick interpolation problem for ${H_2}$ functions. We reduce this problem to a Nevanlinna-Pick interpolation problem for Schur functions and obtain a linear fractional transformation which describes the set of all solutions.

References

D. Alpay and V. Bolotnikov, Two-sided Nevanlinna-Pick interpolation for a class of matrix-valued functions, Z. Anal. Anwendungen 12 (1993), no. 2, 211–238. MR 1245916, DOI 10.4171/ZAA/570
Daniel Alpay, Patrick Dewilde, and Harry Dym, On the existence and construction of solutions to the partial lossless inverse scattering problem with applications to estimation theory, IEEE Trans. Inform. Theory 35 (1989), no. 6, 1184–1205. MR 1036623, DOI 10.1109/18.45275
Daniel Alpay and Harry Dym, On reproducing kernel spaces, the Schur algorithm, and interpolation in a general class of domains, Operator theory and complex analysis (Sapporo, 1991) Oper. Theory Adv. Appl., vol. 59, Birkhäuser, Basel, 1992, pp. 30–77. MR 1246809

De Branges spaces and analytic operator functions

N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.1090/S0002-9947-1950-0051437-7
T. Ya. Azizov and I. S. Iokhvidov, Osnovy teorii lineĭnykh operatorov v prostranstvakh s indefinitnoĭ metrikoĭ, “Nauka”, Moscow, 1986 (Russian). MR 863885
Joseph A. Ball, Israel Gohberg, and Leiba Rodman, Interpolation of rational matrix functions, Operator Theory: Advances and Applications, vol. 45, Birkhäuser Verlag, Basel, 1990. MR 1083145, DOI 10.1007/978-3-0348-7709-1
Joseph A. Ball and J. William Helton, A Beurling-Lax theorem for the Lie group $\textrm {U}(m,\,n)$ which contains most classical interpolation theory, J. Operator Theory 9 (1983), no. 1, 107–142. MR 695942
P. L. Duren and D. L. Williams, Interpolation problems in function spaces, J. Functional Analysis 9 (1972), 75–86. MR 0291787, DOI 10.1016/0022-1236(72)90015-8
Harry Dym, $J$ contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, CBMS Regional Conference Series in Mathematics, vol. 71, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. MR 1004239, DOI 10.1090/cbms/071

A criterion for the solvability of the tangential Nevanlinna-Pick problem

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I. P. Fedčina, A description of the solutions of the Nevanlinna-Pick tangent problem, Akad. Nauk Armjan. SSR Dokl. 60 (1975), no. 1, 37–42 (Russian, with Armenian summary). MR 385104
Dušan R. Georgijević, On operator valued multiple interpolation, Complex analysis and applications ’87 (Varna, 1987) Publ. House Bulgar. Acad. Sci., Sofia, 1989, pp. 176–184. MR 1127632
M. G. Kreĭn and H. Langer, Über die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume $\Pi _{\kappa }$, Hilbert space operators and operator algebras (Proc. Internat. Conf., Tihany, 1970) Colloq. Math. Soc. János Bolyai, vol. 5, North-Holland, Amsterdam, 1972, pp. 353–399 (German). MR 0423122
M. G. Kreĭn and A. A. Nudel′man, The Markov moment problem and extremal problems, Translations of Mathematical Monographs, Vol. 50, American Mathematical Society, Providence, R.I., 1977. Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development; Translated from the Russian by D. Louvish. MR 0458081, DOI 10.1090/mmono/050
Herbert Meschkowski, Hilbertsche Räume mit Kernfunktion, Die Grundlehren der mathematischen Wissenschaften, Band 113, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1962 (German). MR 0140912, DOI 10.1007/978-3-642-94848-0
Stephen Parrott, On a quotient norm and the Sz.-Nagy - Foiaş lifting theorem, J. Functional Analysis 30 (1978), no. 3, 311–328. MR 518338, DOI 10.1016/0022-1236(78)90060-5
Marvin Rosenblum, A corona theorem for countably many functions, Integral Equations Operator Theory 3 (1980), no. 1, 125–137. MR 570865, DOI 10.1007/BF01682874
Marvin Rosenblum and James Rovnyak, Hardy classes and operator theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. Oxford Science Publications. MR 822228
Saburou Saitoh, Theory of reproducing kernels and its applications, Pitman Research Notes in Mathematics Series, vol. 189, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. MR 983117
Donald Sarason, Generalized interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 127 (1967), 179–203. MR 208383, DOI 10.1090/S0002-9947-1967-0208383-8
Donald Sarason, Shift-invariant spaces from the Brangesian point of view, The Bieberbach conjecture (West Lafayette, Ind., 1985) Math. Surveys Monogr., vol. 21, Amer. Math. Soc., Providence, RI, 1986, pp. 153–166. MR 875239, DOI 10.1090/surv/021/13

Exposed points in

Function theory and de Branges spaces

Laurent Schwartz, Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (noyaux reproduisants), J. Analyse Math. 13 (1964), 115–256 (French). MR 179587, DOI 10.1007/BF02786620