Adaptive orthonormal systems for matrix-valued functions

Alpay, Daniel; Colombo, Fabrizio; Qian, Tao; Sabadini, Irene

doi:10.1090/proc/13359

Adaptive orthonormal systems for matrix-valued functions
HTML articles powered by AMS MathViewer

by Daniel Alpay, Fabrizio Colombo, Tao Qian and Irene Sabadini PDF

Proc. Amer. Math. Soc. 145 (2017), 2089-2106 Request permission

Abstract:

In this paper we consider functions in the Hardy space $\mathbf {H}_2^{p\times q}$ defined in the unit disc of matrix-valued functions. We show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably modified matrix-valued Blaschke products, in an adaptive way. The procedure is based on a generalization to the matrix-valued case of the maximum selection principle which involves not only selections of suitable points in the unit disc but also suitable orthogonal projections. We show that the maximum selection principle gives rise to a convergent algorithm. Finally, we discuss the case of real-valued signals.

References

Daniel Alpay, An advanced complex analysis problem book, Birkhäuser/Springer, Cham, 2015. Topological vector spaces, functional analysis, and Hilbert spaces of analytic functions. MR 3410523, DOI 10.1007/978-3-319-16059-7
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, Aad Dijksma, James Rovnyak, and Hendrik de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Operator Theory: Advances and Applications, vol. 96, Birkhäuser Verlag, Basel, 1997. MR 1465432, DOI 10.1007/978-3-0348-8908-7
Daniel Alpay and Harry Dym, On applications of reproducing kernel spaces to the Schur algorithm and rational $J$ unitary factorization, I. Schur methods in operator theory and signal processing, Oper. Theory Adv. Appl., vol. 18, Birkhäuser, Basel, 1986, pp. 89–159. MR 902603, DOI 10.1007/978-3-0348-5483-2_{5}
Daniel Alpay and Israel Gohberg, On orthogonal matrix polynomials, Orthogonal matrix-valued polynomials and applications (Tel Aviv, 1987–88) Oper. Theory Adv. Appl., vol. 34, Birkhäuser, Basel, 1988, pp. 25–46. MR 1021059, DOI 10.1007/978-3-0348-5472-6_{2}
Daniel Alpay and Israel Gohberg, Unitary rational matrix functions, Topics in interpolation theory of rational matrix-valued functions, Oper. Theory Adv. Appl., vol. 33, Birkhäuser, Basel, 1988, pp. 175–222. MR 960699, DOI 10.1007/978-3-0348-5469-6_{5}
Daniel Alpay and H. Turgay Kaptanoğlu, Some finite-dimensional backward-shift-invariant subspaces in the ball and a related interpolation problem, Integral Equations Operator Theory 42 (2002), no. 1, 1–21. MR 1866874, DOI 10.1007/BF01203020
Damir Z. Arov and Harry Dym, $J$-contractive matrix valued functions and related topics, Encyclopedia of Mathematics and its Applications, vol. 116, Cambridge University Press, Cambridge, 2008. MR 2474532, DOI 10.1017/CBO9780511721427
William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
Joseph A. Ball and Vladimir Bolotnikov, Nevanlinna-Pick interpolation for Schur-Agler class functions on domains with matrix polynomial defining function in $\Bbb C^n$, New York J. Math. 11 (2005), 247–290. MR 2154356
Joseph A. Ball, Vladimir Bolotnikov, and Quanlei Fang, Schur-class multipliers on the Arveson space: de Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations, J. Math. Anal. Appl. 341 (2008), no. 1, 519–539. MR 2394102, DOI 10.1016/j.jmaa.2007.10.033
Joseph A. Ball, Tavan T. Trent, and Victor Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, Operator theory and analysis (Amsterdam, 1997) Oper. Theory Adv. Appl., vol. 122, Birkhäuser, Basel, 2001, pp. 89–138. MR 1846055
Harm Bart, Israel Gohberg, and Marinus A. Kaashoek, Minimal factorization of matrix and operator functions, Operator Theory: Advances and Applications, vol. 1, Birkhäuser Verlag, Basel-Boston, Mass., 1979. MR 560504
N. Bourbaki, Topologie générale, Chapitre 5 à 10, Diffusion C.C.L.S., Paris, 1974.
R. R. Coifman and S. Steinerberger, Nonlinear phase unwinding of functions, preprint, arXiv:1508.01241.
Michael Cowling and Tao Qian, A class of singular integrals on the $n$-complex unit sphere, Sci. China Ser. A 42 (1999), no. 12, 1233–1245. MR 1749933, DOI 10.1007/BF02876023
S. W. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), no. 3, 300–304. MR 480362, DOI 10.1090/S0002-9939-1978-0480362-8
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
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Devin C. V. Greene, Stefan Richter, and Carl Sundberg, The structure of inner multipliers on spaces with complete Nevanlinna-Pick kernels, J. Funct. Anal. 194 (2002), no. 2, 311–331. MR 1934606, DOI 10.1006/jfan.2002.3928
M. A. Kaashoek and J. Rovnyak, On the preceding paper by R. B. Leech, Integral Equations Operator Theory 78 (2014), no. 1, 75–77. MR 3147403, DOI 10.1007/s00020-013-2108-7
C. Lance, Hilbert $C^*$-modules: A toolkit for operator algebraists, London Math. Soc. Lecture Notes, 210, 1994.
Peter D. Lax and Ralph S. Phillips, Scattering theory, 2nd ed., Pure and Applied Mathematics, vol. 26, Academic Press, Inc., Boston, MA, 1989. With appendices by Cathleen S. Morawetz and Georg Schmidt. MR 1037774
Robert B. Leech, Factorization of analytic functions and operator inequalities, Integral Equations Operator Theory 78 (2014), no. 1, 71–73. MR 3147402, DOI 10.1007/s00020-013-2107-8
S. Mallat and Z. Zhang, Matching pursuit with time-frequency dictionaries, IEEE Trans. Signal Proc. (1993), 3397–3415.
Wen Mi, Tao Qian, and Feng Wan, A fast adaptive model reduction method based on Takenaka-Malmquist systems, Systems Control Lett. 61 (2012), no. 1, 223–230. MR 2878709, DOI 10.1016/j.sysconle.2011.10.016
V. P. Potapov, The multiplicative structure of $J$-contractive matrix functions, Trudy Moskov. Mat. Obšč. 4 (1955), 125–236 (Russian). MR 0076882
Tao Qian, Cyclic AFD algorithm for the best rational approximation, Math. Methods Appl. Sci. 37 (2014), no. 6, 846–859. MR 3188530, DOI 10.1002/mma.2843
Tao Qian, Adaptive Fourier decompositions and rational approximations, part I: Theory, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014), no. 5, 1461008, 13. MR 3257806, DOI 10.1142/S0219691314610086
Tao Qian, Hong Li, and Michael Stessin, Comparison of adaptive mono-component decompositions, Nonlinear Anal. Real World Appl. 14 (2013), no. 2, 1055–1074. MR 2991133, DOI 10.1016/j.nonrwa.2012.08.017
Tao Qian, Two-dimensional adaptive Fourier decomposition, Math. Methods Appl. Sci. 39 (2016), no. 10, 2431–2448. MR 3512759, DOI 10.1002/mma.3649
Tao Qian and Yan-Bo Wang, Adaptive Fourier series—a variation of greedy algorithm, Adv. Comput. Math. 34 (2011), no. 3, 279–293. MR 2776445, DOI 10.1007/s10444-010-9153-4
Tao Qian, Yan-Bo Wang, and Pei Dang, Adaptive decomposition into mono-components, Adv. Adapt. Data Anal. 1 (2009), no. 4, 703–709. MR 2669206, DOI 10.1142/S1793536909000278
Peter Quiggin, For which reproducing kernel Hilbert spaces is Pick’s theorem true?, Integral Equations Operator Theory 16 (1993), no. 2, 244–266. MR 1205001, DOI 10.1007/BF01358955
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
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
V. N. Temlyakov, Weak greedy algorithms, Adv. Comput. Math. 12 (2000), no. 2-3, 213–227. MR 1745113, DOI 10.1023/A:1018917218956