Constrained Hardy Space Approximation

We consider the problem of minimizing the distance kf − kLp(K), where K is a subset of the complex unit circle @D and ∈ C(K), subject to the constraint that f lies in the Hardy space Hp(D) and |f| ≤ g for some positive function g. This problem occurs in the context of filter design for causal LTI systems. We show that the optimization problem has a unique solution, which satisfies an extremal property similar to that for the Nehari problem. Moreover, we prove that the minimum of the optimization problem can be approximated by smooth functions. This makes the problem accessible for numerical solution, with which we deal in a follow-up paper

Constrained Hardy Space Approximation II: Numerics

We consider the problem of minimizing the distance kf − kLp(K), where K is a subset of the complex unit circle @D and ∈ C(K), subject to the constraint that f lies in the Hardy space Hp(D) and |f| ≤ g for some positive function g. This problem occurs in the context of filter design for causal LTI systems. We show that the optimization problem has a unique solution, which satisfies an extremal property similar to that for the Nehari problem. Moreover, we prove that the minimum of the optimization problem can be approximated by smooth functions. This makes the problem accessible for numerical solution, with which we deal in a follow-up paper

Constrained L2L^2-approximation by polynomials on subsets of the circle

We study best approximation to a given function, in the least square sense on a subset of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary subset. We show that the solution to this problem, as the degree goes large, converges to the solution of a bounded extremal problem for analytic functions which is instrumental in system identification. We provide a numerical example on real data from a hyperfrequency filter

Constrained optimization in classes of analytic functions with prescribed pointwise values

We consider an overdetermined problem for Laplace equation on a disk with partial boundary data where additional pointwise data inside the disk have to be taken into account. After reformulation, this ill-posed problem reduces to a bounded extremal problem of best norm-constrained approximation of partial L2 boundary data by traces of holomorphic functions which satisfy given pointwise interpolation conditions. The problem of best norm-constrained approximation of a given L2 function on a subset of the circle by the trace of a H2 function has been considered in [Baratchart \& Leblond, 1998]. In the present work, we extend such a formulation to the case where the additional interpolation conditions are imposed. We also obtain some new results that can be applied to the original problem: we carry out stability analysis and propose a novel method of evaluation of the approximation and blow-up rates of the solution in terms of a Lagrange parameter leading to a highly-efficient computational algorithm for solving the problem

Effective H^{\infty} interpolation constrained by Hardy and Bergman weighted norms

Given a finite set $\sigma$ of the unit disc $\mathbb{D}$ and a holomorphic function $f$ in $\mathbb{D}$ which belongs to a class $X$ we are looking for a function $g$ in another class $Y$ which minimizes the norm $|g|_{Y}$ among all functions $g$ such that $g_{|\sigma}=f_{|\sigma}$. Generally speaking, the interpolation constant considered is $c(\sigma,\, X,\, Y)={sup}{}_{f\in X,\,\parallel f\parallel_{X}\leq1}{inf}\{|g|_{Y}:\, g_{|\sigma}=f_{|\sigma}\} \,.$ When $Y=H^{\infty}$, our interpolation problem includes those of Nevanlinna-Pick (1916), Caratheodory-Schur (1908). Moreover, Carleson's free interpolation (1958) has also an interpretation in terms of our constant $c(\sigma,\, X,\, H^{\infty})$.} If $X$ is a Hilbert space belonging to the scale of Hardy and Bergman weighted spaces, we show that $c(\sigma,\, X,\, H^{\infty})\leq a\phi_{X}(1-\frac{1-r}{n})$ where n=#\sigma, $r={max}{}_{\lambda\in\sigma}|\lambda|$ and where $\phi_{X}(t)$ stands for the norm of the evaluation functional $f\mapsto f(t)$ on the space $X$. The upper bound is sharp over sets $\sigma$ with given $n$ and $r$.} If $X$ is a general Hardy-Sobolev space or a general weighted Bergman space (not necessarily of Hilbert type), we also found upper and lower bounds for $c(\sigma,\, X,\, H^{\infty})$ (sometimes for special sets $\sigma$) but with some gaps between these bounds.} This constrained interpolation is motivated by some applications in matrix analysis and in operator theory.

Approximation in reflexive Banach spaces and applications to the invariant subspace problem

We formulate a general approximation problem involving reflexive and smooth Banach spaces, and give its explicit solution. Two applications are presented— the first is to the Bounded Completion Problem involving approximation of Hardy class functions, while the second involves the construction of minimal vec- tors and hyperinvariant subspaces of linear operators, generalizing the Hilbert space technique of Ansari and Enflo

Approximation in reflexive Banach spaces and applications to the invariant subspace problem

We formulate a general approximation problem involving reflexive and smooth Banach spaces, and give its explicit solution. Two applications are presented— the first is to the Bounded Completion Problem involving approximation of Hardy class functions, while the second involves the construction of minimal vec- tors and hyperinvariant subspaces of linear operators, generalizing the Hilbert space technique of Ansari and Enflo

Bounds for optimization of the reflection coefficient by constrained optimization in hardy spaces

The purpose of this book is twofold. Our starting point is the design of layered media with a prescribed reflection coefficient. In the first part of this book we show that the space of physically realizable reflection coefficients is rather restricted by a number of properties. In the second part we consider a constrained approximation problem in Hardy spaces. This can be viewed as an optimization problem for the frequency response of a causal LTI system with limited gain

Free biholomorphic functions and operator model theory,II

This is a continuation of the paper entitled "Free biholomorphic functions and operator model theory", in our attempt to transfer the free analogue of Nagy-Foias theory from the unit ball [B(\cH)^n]_1 to other noncommutative domains and varieties in B(\cH)^n, using appropriate maps. The present paper treats the completely non-coisometric case and the case of noncommutative varieties.Comment: 37 page