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Adaptive orthonormal systems for matrix-valued functions


Authors: Daniel Alpay, Fabrizio Colombo, Tao Qian and Irene Sabadini
Journal: Proc. Amer. Math. Soc. 145 (2017), 2089-2106
MSC (2010): Primary 47A56, 41A20, 30H10
DOI: https://doi.org/10.1090/proc/13359
Published electronically: January 6, 2017
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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.


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Additional Information

Daniel Alpay
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Email: dany@math.bgu.ac.il

Fabrizio Colombo
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, 20133 Milano, Italy
Email: fabrizio.colombo@polimi.it

Tao Qian
Affiliation: Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau
Email: fsttq@umac.mo

Irene Sabadini
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, 20133 Milano, Italy
Email: irene.sabadini@polimi.it

DOI: https://doi.org/10.1090/proc/13359
Keywords: Matrix-valued functions and Hardy spaces, matrix-valued Blaschke products, maximum selection principle, adaptive decomposition
Received by editor(s): May 1, 2016
Received by editor(s) in revised form: June 30, 2016
Published electronically: January 6, 2017
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2017 American Mathematical Society