On frames that are iterates of a multiplication operator

• Aydin Shukurov ^[1] ; Afet Jabrailova ^[1]
  1. [1] Institute of Mathematics and Mechanics

     Institute of Mathematics and Mechanics

     Azerbaiyán

     
• Localización: Revista Colombiana de Matemáticas, ISSN-e 0034-7426, Vol. 55, Nº. 2, 2021, págs. 139-147
• Idioma: inglés
• DOI: 10.15446/recolma.v55n2.102513
• Títulos paralelos:
  • Sobre marcos que son iteraciones de un operador de multiplicación
• Enlaces
  • Texto completo
• Resumen
  • español

    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.

  • English

    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=-∞.

• Referencias bibliográficas
  • A. Aldroubi, C. Cabrelli, A. F. Cakmak, U. Molter, and A. Petrosyan, Iterative actions of normal operators, J. Funct. Anal. 272 (2017), no....
  • A. Aldroubi, C. Cabrelli, U. Molter, and Tang S., Dynamical sampling, Appl. Harm. Anal. Appl. 42 (2017), no. 3, 378-401. DOI: https://doi.org/10.1016/j.acha.2015.08.014
  • A. Aldroubi and A. Petrosyan, Dynamical sampling and systems from iterative actions of operators, Preprint, 2016. DOI: https://doi.org/10.1007/978-3-319-55550-8_2
  • 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. DOI: https://doi.org/10.1007/978-3-319-25613-9
  • O. Christensen and M. Hasannasab, Frame properties of systems arising via iterative actions of operators, Preprint, 2016.
  • O. Christensen and M. Hasannasab, Operator representations of frames: boundedness, duality, and stability, Integral Equations and Operator...
  • O. Christensen, M. Hasannasab, and F. Philipp, Frame Properties of Operator Orbits, arXiv preprint arXiv:1804.03438, 2018. DOI: https://doi.org/10.1002/mana.201800344
  • O. Christensen, M. Hasannasab, and D. Stoeva, Operator representations of sequences and dynamical sampling, arXiv preprint arXiv:1804.00077,...
  • I. M. Gelfand, Several problems on the theory of functions of real variable; Problem 17, Uspekhi Mat. Nauk (1938), no. 5, 233.
  • I. P. Natanson, Theory of Functions of a Real Variable, Nauka, Moscow, 1974.
  • Christensen O, M. Hasannasab, and E. Rashidi, Dynamical sampling and frame representations with bounded operators, J. Math. Anal. Appl. 463...
  • F. Philipp, Bessel orbits of normal operators, J. Math. Anal. Appl. 448 (2017), no. 2, 767-785. DOI: https://doi.org/10.1016/j.jmaa.2016.11.009
  • A. Sh. Shukurov, Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces, Colloq. Math. 127 (2012), no. 1, 105-109....
  • A. Sh. Shukurov, Addendum to “Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces”, Colloq. Math. 137 (2014),...
  • A. Sh. Shukurov, The power system is never a basis in the space of continuous functions, Amer. Math. Monthly 122 (2015), no. 2, 137. DOI:...
  • A. Sh. Shukurov, Impossibility of power series expansion for continuous functions, Azerb. J. Math. 6 (2016), no. 1, 122-125.
  • A. Sh. Shukurov and Z. A. Kasumov, On frame properties of iterates of a multiplication operator, Results Math. 74 (2019), no. 2, 74-84. DOI:...
  • I. Singer, Bases in banach spaces ii, Springer, 1981. DOI: https://doi.org/10.1007/978-3-642-67844-8
  • R. M. Young, An introduction to nonharmonic Fourier series, Academic Press, 1980.