School of Mathematics, Statistics and Physics

Seminar Item

Pure Maths (Analysis) - Dimitris Chiotis (Newcastle)

A superoptimal approximation algorithm based on exterior products

Date/Time: Monday 9 November, 3.00pm - Analysis Seminar

Venue: Zoom


Throughout this talk, we let

• L∞(T, C m×n ) be the space of essentially bounded Lebesgue measurable matrixvalued functions on the unit circle T with the essential supremum norm;

• H∞(D, C m×n ) be the space of bounded analytic matrix-valued functions on the unit disc D with the supremum norm;

• C(T, C m×n ) be the space of continuous matrix-valued functions.

The problem we study is the following.

Superoptimal analytic approximation problem. Given a matrix-valued function G ∈ H∞(D, C m×n ) + C(T, C m×n ), find its superoptimal approximant, that is, a matrix-valued function Q ∈ H∞(D, C m×n ) such that the sequence

s ∞(G − Q) = (s ∞ 0 (G − Q), s∞ 1 (G − Q), s∞ 2 (G − Q), . . .)

is lexicographically minimized over all bounded analytic matrix-valued functions. Here s∞ j (G − Q) = ess supz∈T sj (G(z) − Q(z)) and sj (G(z) − Q(z)) denotes the j-th singular value of the matrix G(z) − Q(z).

We present our construction of an algorithm that utilises exterior products of operators to determine the superoptimal approximant for a given matrix-valued function. The talk is based on joint work with Z.A. Lykova and N.J. Young.