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2, 244–266. [33] I. V. P. Potapov, S even Papers Translated from the Russian, Amer. Math. Soc. Transl. , 1988. ¨ [34] R. Nevanlinna, Uber beschr¨ ankte Funktionen, Ann. Acad. Sci. Fenn. Ser. A 32 (1939), no. 7. 164 Ball and Bolotnikov IEOT [35] M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Oxford University Press, 1985. [36] W. Rudin, F unction theory in the unit ball of Cn , Springer-Verlag, New York, 1980. [37] D. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, University of Arkansas Lecture Notes in the Matehmatical Sciences, Wiley, 1994.

11) 156 Ball and Bolotnikov IEOT 3. 12) for some Schur function S ∈ Sd (E, E∗ ⊕ (⊕d1 E)): S(z) = E∗ S0 (z) : E→ , S1 (z) ⊕d1 E Z(z) = z1 IE ... zd IE . 13) Proof: The equivalence (1 ⇔ 2) follows from a more general fact that F is a contractive multiplier between two reproducing kernel Hilbert spaces H(K1 ) and H(K2 ) of functions analytic on a set Ω if and only if the kernel K2 (z, w) − F (z)K1 (z, w)F (w)∗ is positive on Ω. To show that (2 ⇔ 3), we represent KF as KF (z, w) = IE − F (z)F (w)∗ + F (z)Z(z)Z(w)∗ F (w)∗ , 1 − z, w or, equivalently, as KF (z, w) = A(z)A(w)∗ − B(z)B(w)∗ , 1 − z, w where A(z) = IE F (z)Z(z) and B(z) = F (z).

To appear. Vol. 46 (2003) Bitangential Interpolation 163 [18] V. Bolotnikov and H. Dym, On degenerate interpolation, entropy and extremal problems for matrix Schur functions, Integral Equations Operator Theory, 32 (1998), No. 4, 367–435. [19] V. Bolotnikov and H. Dym, On boundary interpolation for matrix Schur functions, Preprint MCS99-22, Department of Mathematics, The Weizmann Institute of Science, Israel. I. V. Gusev and A. Lindquist, From finite covariance windows to modeling filters: a convex optimization approach, SIAM Review 43(4) (2001), 645-675.

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