Schur-Agler class rational inner functions on the tridisk
HTML articles powered by AMS MathViewer
- by Greg Knese PDF
- 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
Additional Information
- Greg Knese
- Affiliation: Department of Mathematics, Box 870350, University of Alabama, Tuscaloosa, Alabama 35487-0350
- MR Author ID: 813491
- Email: geknese@bama.ua.edu
- Received by editor(s): October 5, 2010
- Published electronically: March 30, 2011
- Additional Notes: This research was supported by NSF grant DMS-1048775
- Communicated by: Richard Rochberg
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 4063-4072
- MSC (2010): Primary 47A57; Secondary 42B05
- DOI: https://doi.org/10.1090/S0002-9939-2011-10975-4
- MathSciNet review: 2823051