Publicado

2022-05-18

On frames that are iterates of a multiplication operator

Sobre marcos que son iteraciones de un operador de multiplicación

Palabras clave:

Dynamical sampling, Operator orbit, frame, Schauder bases, system of powers, Lebesgue spaces (en)
Muestreo dinámico, marcos,, órbitas de operadores, bases de Schauder, sistema de potencias, espacios de Lebesgue (es)

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Autores/as

  • Aydin Shukurov Institute of Mathematics and Mechanics
  • Afet Jabrailova Institute of Mathematics and Mechanics

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=-.

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 {φn}n=0 no puede ser un marco para L2(a, b). La situación cambia radicalmente cuando se consideran sistemas de la forma {φn}n=- en vez de {φn}n=0. El objetivo de este artículo es caracterizar marcos de la forma {φn}∞n=-∞ que son iteraciones del operador multiplicativo Tφ. En esta nota probamos que el problema se reduce al siguiente:

Problema. Caracterice la clase de funciones α para las cuales {einα(·)}+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 {einα(·)}+n=- es una base de Schauder y de Riesz e inclusive cuándo es una base ortonormal.

Referencias

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A. Aldroubi, C. Cabrelli, U. Molter, and Tang S., Dynamical sampling, Appl. Harm. Anal. Appl. 42 (2017), no. 3, 378-401.

A. Aldroubi and A. Petrosyan, Dynamical sampling and systems from iterative actions of operators, Preprint, 2016.

M. S. Brodskii, On a problem of I. M. Gelfand, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 2(74), 129-132.

C. Cabrelli, U. Molter, V. Paternostro, and F. Philipp, Dynamical sampling on finite index sets, Preprint, 2017.

O. Christensen, An introduction to frames and Riesz bases, Second expanded edition. Birkhäuser, 2016.

O. Christensen and M. Hasannasab, Frame properties of systems arising via iterative actions of operators, Preprint, 2016.

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O. Christensen, M. Hasannasab, and F. Philipp, Frame Properties of Operator Orbits, arXiv preprint arXiv:1804.03438, 2018.

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Cómo citar

APA

Shukurov, A., & Jabrailova, A. (2022). On frames that are iterates of a multiplication operator. Revista Colombiana de Matemáticas, 55(2), 139–147. Recuperado a partir de https://revistas.unal.edu.co/index.php/recolma/article/view/102513

ACM

[1]
Shukurov, A. y Jabrailova, A. 2022. On frames that are iterates of a multiplication operator. Revista Colombiana de Matemáticas. 55, 2 (may 2022), 139–147.

ACS

(1)
Shukurov, A.; Jabrailova, A. On frames that are iterates of a multiplication operator. rev.colomb.mat 2022, 55, 139-147.

ABNT

SHUKUROV, A.; JABRAILOVA, A. On frames that are iterates of a multiplication operator. Revista Colombiana de Matemáticas, [S. l.], v. 55, n. 2, p. 139–147, 2022. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/102513. Acesso em: 16 ago. 2022.

Chicago

Shukurov, Aydin, y Afet Jabrailova. 2022. «On frames that are iterates of a multiplication operator». Revista Colombiana De Matemáticas 55 (2):139-47. https://revistas.unal.edu.co/index.php/recolma/article/view/102513.

Harvard

Shukurov, A. y Jabrailova, A. (2022) «On frames that are iterates of a multiplication operator», Revista Colombiana de Matemáticas, 55(2), pp. 139–147. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/102513 (Accedido: 16agosto2022).

IEEE

[1]
A. Shukurov y A. Jabrailova, «On frames that are iterates of a multiplication operator», rev.colomb.mat, vol. 55, n.º 2, pp. 139–147, may 2022.

MLA

Shukurov, A., y A. Jabrailova. «On frames that are iterates of a multiplication operator». Revista Colombiana de Matemáticas, vol. 55, n.º 2, mayo de 2022, pp. 139-47, https://revistas.unal.edu.co/index.php/recolma/article/view/102513.

Turabian

Shukurov, Aydin, y Afet Jabrailova. «On frames that are iterates of a multiplication operator». Revista Colombiana de Matemáticas 55, no. 2 (mayo 18, 2022): 139–147. Accedido agosto 16, 2022. https://revistas.unal.edu.co/index.php/recolma/article/view/102513.

Vancouver

1.
Shukurov A, Jabrailova A. On frames that are iterates of a multiplication operator. rev.colomb.mat [Internet]. 18 de mayo de 2022 [citado 16 de agosto de 2022];55(2):139-47. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/102513

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