Schur-Agler class rational inner functions on the tridisk

Knese, Greg

doi:10.1090/S0002-9939-2011-10975-4

Schur-Agler class rational inner functions on the tridisk
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Proc. Amer. Math. Soc. 139 (2011), 4063-4072 Request permission

Abstract:

We prove two results with regard to rational inner functions in the Schur-Agler class of the tridisk. Every rational inner function of degree $(n,1,1)$ is in the Schur-Agler class, and every rational inner function of degree $(n,m,1)$ is in the Schur-Agler class after multiplication by a monomial of sufficiently high degree.

References

Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR 1882259, DOI 10.1090/gsm/044
J. Milne Anderson, Michael A. Dritschel, and James Rovnyak, Schwarz-Pick inequalities for the Schur-Agler class on the polydisk and unit ball, Comput. Methods Funct. Theory 8 (2008), no. 1-2, 339–361. MR 2419482, DOI 10.1007/BF03321692
Mihály Bakonyi and Geir Naevdal, On the matrix completion method for multidimensional moment problems, Acta Sci. Math. (Szeged) 64 (1998), no. 3-4, 547–558. MR 1666043
Joseph A. Ball and Vladimir Bolotnikov, A tangential interpolation problem on the distinguished boundary of the polydisk for the Schur-Agler class, J. Math. Anal. Appl. 273 (2002), no. 2, 328–348. MR 1932492, DOI 10.1016/S0022-247X(02)00226-3
Joseph A. Ball and Vladimir Bolotnikov, Canonical de Branges-Rovnyak model transfer-function realization for multivariable Schur-class functions, Hilbert spaces of analytic functions, CRM Proc. Lecture Notes, vol. 51, Amer. Math. Soc., Providence, RI, 2010, pp. 1–39. MR 2648864, DOI 10.1090/crmp/051/01
Joseph A. Ball, Cora Sadosky, and Victor Vinnikov, Scattering systems with several evolutions and multidimensional input/state/output systems, Integral Equations Operator Theory 52 (2005), no. 3, 323–393. MR 2184571, DOI 10.1007/s00020-005-1351-y
Brian J. Cole and John Wermer, Ando’s theorem and sums of squares, Indiana Univ. Math. J. 48 (1999), no. 3, 767–791. MR 1736979, DOI 10.1512/iumj.1999.48.1716
M. J. Crabb and A. M. Davie, von Neumann’s inequality for Hilbert space operators, Bull. London Math. Soc. 7 (1975), 49–50. MR 365179, DOI 10.1112/blms/7.1.49
Michael A. Dritschel, On factorization of trigonometric polynomials, Integral Equations Operator Theory 49 (2004), no. 1, 11–42. MR 2057766, DOI 10.1007/s00020-002-1198-4
Bogdan Dumitrescu, Positive trigonometric polynomials and signal processing applications, Signals and Communication Technology, Springer, Dordrecht, 2007. MR 2309555
Jean-Pierre Gabardo, Trigonometric moment problems for arbitrary finite subsets of $\textbf {Z}^n$, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4473–4498. MR 1443194, DOI 10.1090/S0002-9947-98-02091-1
Jeffrey S. Geronimo and Ming-Jun Lai, Factorization of multivariate positive Laurent polynomials, J. Approx. Theory 139 (2006), no. 1-2, 327–345. MR 2220044, DOI 10.1016/j.jat.2005.09.010
Jeffrey S. Geronimo and Hugo J. Woerdeman, Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables, Ann. of Math. (2) 160 (2004), no. 3, 839–906. MR 2144970, DOI 10.4007/annals.2004.160.839
John A. Holbrook, Schur norms and the multivariate von Neumann inequality, Recent advances in operator theory and related topics (Szeged, 1999) Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 375–386. MR 1902811
Greg Knese, Bernstein-Szegő measures on the two dimensional torus, Indiana Univ. Math. J. 57 (2008), no. 3, 1353–1376. MR 2429095, DOI 10.1512/iumj.2008.57.3226
Greg Knese, Polynomials with no zeros on the bidisk, Anal. PDE 3 (2010), no. 2, 109–149. MR 2657451, DOI 10.2140/apde.2010.3.109
Knese, G. (2010b). Rational inner functions in the Schur-Agler class of the polydisk. To appear in Publicacions Matemàtiques.
Knese, G. (2010c). Stable symmetric polynomials and the Schur-Agler class. Preprint.
Anton Kummert, Synthesis of $3$-D lossless first-order one ports with lumped elements, IEEE Trans. Circuits and Systems 36 (1989), no. 11, 1445–1449. MR 1020132, DOI 10.1109/31.41302
Anton Kummert, Synthesis of two-dimensional lossless $m$-ports with prescribed scattering matrix, Circuits Systems Signal Process. 8 (1989), no. 1, 97–119. MR 998029, DOI 10.1007/BF01598747
B. A. Lotto, von Neumann’s inequality for commuting, diagonalizable contractions. I, Proc. Amer. Math. Soc. 120 (1994), no. 3, 889–895. MR 1169881, DOI 10.1090/S0002-9939-1994-1169881-8
Megretski, A. (2003). Positivity of trigonometric polynomials. In Decision and Control, 2003. Proceedings of 42nd IEEE Conference on Decision and Control, volume 4, IEEE, pages 3814 – 3817.
Daniel G. Quillen, On the representation of hermitian forms as sums of squares, Invent. Math. 5 (1968), 237–242. MR 233770, DOI 10.1007/BF01389773
Walter Rudin, The extension problem for positive-definite functions, Illinois J. Math. 7 (1963), 532–539. MR 151796
Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
N. Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Functional Analysis 16 (1974), 83–100. MR 0355642, DOI 10.1016/0022-1236(74)90071-8