TY - JOUR

T1 - Generalized interpolation in H∞ with a complexity constraint

AU - Byrnes, Christopher I.

AU - Georgiou, Tryphon T.

AU - Lindquist, Anders

AU - Megretski, Alexander

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2006/3

Y1 - 2006/3

N2 - In a seminal paper, Sarason generalized some classical interpolation problems for H∞ functions on the unit disc to problems concerning lifting onto H2 of an operator T that is defined on K = H2⊖φH2 (φ is an inner function) and commutes with the (compressed) shift 5. In particular, he showed that interpolants (i.e., f ∈ H∞ such that f(S) = T) having norm equal to ∥T∥ exist, and that in certain cases such an / is unique and can be expressed as a fraction f = b/a with a, 6 ∈ K. In this paper, we study interpolants that are such fractions of K functions and are bounded in norm by 1 (assuming that ∥T∥ < 1, in which case they always exist). We parameterize the collection of all such pairs (a, b) ∈ K × K and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where 0 is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

AB - In a seminal paper, Sarason generalized some classical interpolation problems for H∞ functions on the unit disc to problems concerning lifting onto H2 of an operator T that is defined on K = H2⊖φH2 (φ is an inner function) and commutes with the (compressed) shift 5. In particular, he showed that interpolants (i.e., f ∈ H∞ such that f(S) = T) having norm equal to ∥T∥ exist, and that in certain cases such an / is unique and can be expressed as a fraction f = b/a with a, 6 ∈ K. In this paper, we study interpolants that are such fractions of K functions and are bounded in norm by 1 (assuming that ∥T∥ < 1, in which case they always exist). We parameterize the collection of all such pairs (a, b) ∈ K × K and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where 0 is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

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U2 - 10.1090/S0002-9947-04-03616-5

DO - 10.1090/S0002-9947-04-03616-5

M3 - Article

AN - SCOPUS:33244480238

VL - 358

SP - 965

EP - 987

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 3

ER -