The Power System is Never a Basis in the Space of Continuous Functions

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=-∞.

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Basis of the Properties of Weighted Exponential Systems With Excess

Shukurov¹

2018

Vestnik of Samara University. Natural Science Series

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The main aim of this paper is the determination of a class of such functions for which a weighted exponential system becomes complete and minimal in appropriate space when exactly one of its terms is eliminated. It is shown that the system, obtained in this way cannot be a Schouder basis in this space. The last fact shows that Muckenhoupt-type criterion for the exponential system to be the Schauder basis in Lebesgue spaces after elimination of an element does not exist. This paper generalizes the results of the paper by E.S. Golubeva.

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The Power System is Never a Basis in the Space of Continuous Functions

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On Frame Properties of Iterates of a Multiplication Operator

On Frame Properties of Iterates of a Multiplication Operator

On frames that are iterates of a multiplication operator

Basis of the Properties of Weighted Exponential Systems With Excess

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