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Abstract:

In this article we study invariance properties of shift-invariant spaces in higher dimensions. We state and prove several necessary and sufficient conditions for a shift-invariant space to be invariant under a given closed subgroup of ℝ, and prove the existence of shift-invariant spaces that are exactly invariant for each given subgroup. As an application we relate the extra invariance to the size of support of the Fourier transform of the generators of the shift-invariant space. This work extends recent results obtained for the case of one variable to several variables. © 2010 Birkhäuser Verlag Basel/Switzerland.

Registro:

Documento: Artículo
Título:Invariance of a Shift-Invariant Space in Several Variables
Autor:Anastasio, M.; Cabrelli, C.; Paternostro, V.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina
CONICET, Consejo Nacional de Investigaciones Científicas y Técnicas, Buenos Aires, Argentina
Palabras clave:Fibers; Range functions; Shift-invariant space; Translation invariant space
Año:2011
Volumen:5
Número:4
Página de inicio:1031
Página de fin:1050
DOI: http://dx.doi.org/10.1007/s11785-010-0045-x
Título revista:Complex Analysis and Operator Theory
Título revista abreviado:Complex Anal. Oper. Theory
ISSN:16618254
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_16618254_v5_n4_p1031_Anastasio

Referencias:

  • Aldroubi, A., Cabrelli, C., Heil, C., Kornelson, K., Molter, U., Invariance of a Shift-Invariant Space (2010) J. Fourier Anal. Appl., 16 (1), pp. 60-75
  • de Boor, C., Devore, R.A., Ron, A., The structure of finitely generated shift-invariant spaces in L2(ℝ2 (1994) J. Funct. Anal., 119 (1), pp. 37-78
  • de Boor, C., Devore, R.A., Ron, A., Approximation from shift-invariant subspaces of L2(ℝ2 (1994) Trans. Am. Math. Soc., 341 (2), pp. 787-806
  • Bownik, M., Kaiblinger, N., Minimal generator sets for finitely generated shift-invariant subspaces of L2(ℝn (2006) J. Math. Anal. Appl., 313 (1), pp. 342-352
  • Bourbaki, N., (1974) Éléments de mathématique. Topologie générale, 10, p. 336. , Chapitres 5 à,Springer, Berlin. Reprint of the 1974 original
  • Bourbaki, N., (1981) Éléments de mathématique. Algèbre, 7 VII, p. 422. , Chapitres 4 à,Masson, Paris
  • Bownik, M., The structure of shift-invariant subspaces of L2(ℝd (2000) J. Funct. Anal., 177 (2), pp. 282-309
  • Helson, H., (1964) Lectures on Invariant Subspaces, XI, p. 130. , Academic Press, New York
  • Hogan, J.A., Lakey, J.D., Non-translation-invariance in principal shift-invariant spaces (2005) Advances in Analysis. Proceedings of the 4th International ISAAC Congress, pp. 471-485. , In: Begehr, H. G. W., et al. (ed.),Toronto, Canada, August 11-16, 2003,World Scientific, Hackensack
  • Hewitt, E., Ross, K.A., (1963) Abstract harmonic analysis, I. , Structure of topological groups. Integration theory. Group representations, Die Grundlehren der mathematischen Wissenschaften. 115. Springer, Berlin
  • Ron, A., Shen, Z., Frames and stable bases for shift-invariant subspaces of L<sup/>(ℝd (1995) Can. J. Math., 47 (5), pp. 1051-1094
  • Srinivasan, T.P., Doubly invariant subspaces (1964) Pac. J. Math., 14, pp. 701-707

Citas:

---------- APA ----------
Anastasio, M., Cabrelli, C. & Paternostro, V. (2011) . Invariance of a Shift-Invariant Space in Several Variables. Complex Analysis and Operator Theory, 5(4), 1031-1050.
http://dx.doi.org/10.1007/s11785-010-0045-x
---------- CHICAGO ----------
Anastasio, M., Cabrelli, C., Paternostro, V. "Invariance of a Shift-Invariant Space in Several Variables" . Complex Analysis and Operator Theory 5, no. 4 (2011) : 1031-1050.
http://dx.doi.org/10.1007/s11785-010-0045-x
---------- MLA ----------
Anastasio, M., Cabrelli, C., Paternostro, V. "Invariance of a Shift-Invariant Space in Several Variables" . Complex Analysis and Operator Theory, vol. 5, no. 4, 2011, pp. 1031-1050.
http://dx.doi.org/10.1007/s11785-010-0045-x
---------- VANCOUVER ----------
Anastasio, M., Cabrelli, C., Paternostro, V. Invariance of a Shift-Invariant Space in Several Variables. Complex Anal. Oper. Theory. 2011;5(4):1031-1050.
http://dx.doi.org/10.1007/s11785-010-0045-x