On frames that are iterates of a multiplication operator

Abstract

A result from the recent paper of the first named author on frame properties of iterates of the multiplication operator Tφf = φf implies in particular that a system of the form {φn}∞n=0 cannot be a frame in L2(a, b). The classical exponential system shows that the situation changes drastically when one considers systems of the form {φn}∞n=-∞ instead of {φn}∞n=0. This note is dedicated to the characterization of all frames of the form {φn}∞n=-∞ coming from iterates of the multiplication operator Tφ. It is shown in this note that this problem can be reduced to the following one: Problem. Find (or describe a class of ) all real-valued functions α for which {einα(·)}+∞n=-∞ is a frame in L2(a, b). In this note we give a partial answer to this problem. To our knowledge, in the general statement, this problem remains unanswered not only for frame, but also for Schauder and Riesz basicity properties and even for orthonormal basicity of systems of the form {einα(·)}+∞n=-∞.