Approximation of infinite matrices by matricial Haar polynomials

Abstract

The main goal of this paper is to extend the approximation theorem of contiuous functions by Haar polynomials (see Theorem A) to infinite matrices (see Theorem C). The extension to the matricial framework will be based on the one hand on the remark that periodic functions which belong to L^∞ (T) may be one-to-one identified with Toeplitz matrices from B(l₂) (see Theorem 0) and on the other hand on some notions given in the paper. We mention for instance: ms—a unital commutative subalgebra of l^∞, C(l₂) the matricial analogue of the space of all continuous periodic functions C(T), the matricial Haar polynomials, etc.

In Section 1 we present some results concerning the space ms—a concept important for this generalization, the proof of the main theorem being given in the second section.