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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.55 no.2 Bogotá July/Dec. 2021 Epub May 31, 2022
https://doi.org/10.15446/recolma.v55n2.102513
ORIGINAL ARTICLES
On frames that are iterates of a multiplication operator
Sobre marcos que son iteraciones de un operador de multiplicación
1Institute of Mathematics and Mechanics, Baku, Azerbaijan
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 {φ n }∞ n=0 cannot be a frame in L 2(a, b). The classical exponential system shows that the situation changes drastically when one considers systems of the form {φ n }∞ n=- ∞ instead of {φ n }∞ n= 0. This note is dedicated to the characterization of all frames of the form {φ n }∞ n=- ∞ coming from iterates of the multiplication operator T (. 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 {e inα(.) }+∞ 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 {e inα(.) }+∞ n=-∞ .
Keywords: Dynamical sampling; Operator orbit; frame; Schauder bases; system of powers; Lebesgue spaces
Un resultado reciente por parte del primer autor del artículo acerca de marcos muestra que para las iteraciones del operador multiplicativo Tφf = φf un sistema de la forma {φ n }∞ n=0 no puede ser un marco para L 2(a, b). La situación cambia radicalmente cuando se consideran sistemas de la forma {φ n }∞ n=- ∞ en vez de {φ n }∞ n=0 . El objetivo de este artículo es caracterizar marcos de la forma {φ n }( n=- ∞ que son iteraciones del operador multiplicativo T φ. En esta nota probamos que el problema se reduce al siguiente:
Problema. Caracterice la clase de funciones ( para las cuales {e inα(.) }∞ n=- ∞ es un marco de L2(a, b).
En este artículo damos una respuesta parcial al problema. Hasta donde sabemos, en el caso general el problema sigue abierto, no sólo para marcos, sino también para determinar cuándo la familia {e inα(.) }+∞ n=- ∞ es una base de Schauder y de Riesz e inclusive cuándo es una base ortonormal.
Palabras clave: Muestreo dinámico; marcos; órbitas de operadores; bases de Schauder; sistema de potencias; espacios de Lebesgue
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Received: April 06, 2020; Accepted: August 05, 2020
This is an open-access article distributed under the terms of the Creative Commons Attribution License
