Generalized interpolation in 𝐻^{∞} with a complexity constraint

Byrnes, Christopher; Georgiou, Tryphon; Lindquist, Anders; Megretski, Alexander

doi:10.1090/S0002-9947-04-03616-5

Generalized interpolation in $H^\infty$ with a complexity constraint
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by Christopher I. Byrnes, Tryphon T. Georgiou, Anders Lindquist and Alexander Megretski PDF

Trans. Amer. Math. Soc. 358 (2006), 965-987 Request permission

Abstract:

In a seminal paper, Sarason generalized some classical interpolation problems for $H^\infty$ functions on the unit disc to problems concerning lifting onto $H^2$ of an operator $T$ that is defined on $\mathcal {K} =H^2\ominus \phi H^2$ ($\phi$ is an inner function) and commutes with the (compressed) shift $S$. In particular, he showed that interpolants (i.e., $f\in H^\infty$ such that $f(S)=T$) having norm equal to $\|T\|$ exist, and that in certain cases such an $f$ is unique and can be expressed as a fraction $f=b/a$ with $a,b\in \mathcal {K}$. In this paper, we study interpolants that are such fractions of $\mathcal {K}$ 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 $(a,b)\in \mathcal {K}\times \mathcal {K}$ 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 $\phi$ is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

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