Best Approximations in Hardy Spaces on Infinite-Dimensional Unitary Matrix Groups

Oleh Lopushansky

doi:10.1155/2014/631503

Abstract

We investigate the problem of best approximations in the Hardy space of complex functions, defined on the infinite-dimensional unitary matrix group. Applying an abstract Besov-type interpolation scale and the Bernstein-Jackson inequalities, a behavior of such approximations is described. An application to best approximations in symmetric Fock spaces is shown.

Highlights

We investigate the problem of best approximations in the Hardy space of complex functions, defined on the infinite-dimensional unitary matrix group
Our goal is to investigate a best approximation problem in the quasinormed Hardy space Hχp (0 < p ≤ ∞) of complex functions of infinitely many variables
The considered Hardy space is defined on the infinite-dimensional unitary matrix groups U(∞), acting over a suitable infinite-dimensional Hilbert space E

Summary

Introduction

Our goal is to investigate a best approximation problem in the quasinormed Hardy space Hχp (0 < p ≤ ∞) of complex functions of infinitely many variables. To solve this problem, we use an interpolating scale of special Besov-type subgroups Bτα(G), defined by approximation Efunctionals. The main result is in Theorem 7 that, in some sense, gives a solution of best approximation problem in the Hardy spaces Hχp for the case of Besov-type scale. In Theorem 8 we show an application of the Bernstein-Jackson inequalities to best linear and nonlinear approximations in symmetric Fock spaces

Besov-Type Approximation Scales

Best Approximations

Hardy Spaces of Infinitely Many Variables

Approximations in Hardy Spaces

Applications to Symmetric Fock Spaces