Navegar

Colección

Datos Estadísticas

Artículo

Este artículo es de Acceso Abierto y puede ser descargado en su versión final desde nuestro repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

We prove the existence of Lpfunctions satisfying a kind of self-similarity condition. This is achieved by solving a functional equation by means of the construction of a contractive operator on an appropriate functional space. The solution, a fixed point of the operator, can be obtained by an iterative process, making this model very suitable to use in applications such as fractal image and signal compression. On the other hand, this "generalized self-similarity equation" includes matrix refinement equations of the typef(x)=∑ckf(Ax-k) which are central in the construction of wavelets and multiwavelets. The results of this paper will therefore yield conditions for the existence of Lp-refinable functions in a very general setting. © 1998 Academic Press.

Registro:

Documento: Artículo
Título:Generalized Self-Similarity
Autor:Cabrelli, C.A.; Molter, U.M.
Filiación:Departamento de Matemática, Fac. de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, 1428, Capital Federal, Argentina
Palabras clave:Dilation equation; Fixed points; Fractals; Functional equation; Inverse problem for fractals; Refinement equation; Self-similarity; Wavelets
Año:1999
Volumen:230
Número:1
Página de inicio:251
Página de fin:260
DOI: http://dx.doi.org/10.1006/jmaa.1998.6200
Título revista:Journal of Mathematical Analysis and Applications
Título revista abreviado:J. Math. Anal. Appl.
ISSN:0022247X
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_0022247X_v230_n1_p251_Cabrelli.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v230_n1_p251_Cabrelli

Referencias:

  • Abenda, S., Turchetti, G., Inverse problem of fractal sets on the real line via the moment method (1989) Nuovo Cimento, 104, pp. 213-227
  • Bajraktarevic, M., Sur une équation fonctionnelle (1957) Glasnik Mat.-Fiz. I Astr., 12, pp. 201-205
  • Barnsley, M.F., Fractal functions and interpolation (1986) Constr. Approx., 2, pp. 303-329
  • Barnsley, M.F., (1988) Fractals Everywhere, , San Diego: Academic Press
  • Barnsley, M.F., Ervin, V., Hardin, D., Lancaster, J., Solution of an inverse problem for fractals and other sets (1985) Proceedings of the National Academy of Sciences, , p. 1975-1977
  • Barnsley, M.F., Harrington, A.N., The calculus of fractal interpolation functions (1989) J. Approx. Theory, 57, pp. 14-34
  • Barnsley, M.F., Sloan, A.D., A better way to compress images (1988) BYTE Mag., pp. 215-223
  • Cabrelli, C.A., Falsetti, M.C., Molter, U.M., Block-coding (1998) A Functional Approach
  • Cabrelli, C.A., Forte, B., Molter, U.M., Vrscay, E.R., Iterated fuzzy set systems: A new approach to the inverse problem for fractals and other sets (1992) J. Math. Anal. Appl., 171, pp. 79-100
  • Cabrelli, C.A., Heil, C., Molter, U.M., Generalized self-similarity applied to matrix dilation equations, (extended abstract) (1996) Z. Angew. Math. Mech., 76, pp. 493-494
  • Cabrelli, C.A., Heil, C., Molter, U.M., (1998) Self-similarity and Multiwavelets in Higher Dimensions
  • Cabrelli, C.A., Molter, U.M., Density of fuzzy attractors: A step towards the solution of the inverse problem for fractals and other sets (1991) Proceedings of the NATO ASI on Probabilistic and Stochastic Methods in Analysis with Applications, pp. 163-173. , J. S. Byrnes
  • De Rahm, G., Sur quelques courbes définies par des équations fonctionnelles (1957) Rend. Sem. Mat. Univ. Politec. Torino, 16, pp. 101-113
  • Deliu, A., Geronimo, J., Schonkwiler, R., On the inverse problem for two-dimensional attractors (1997) Philos. Trans. R. Soc. Lond., Ser. A, 355, pp. 1017-1062
  • Dubuc, S., Functional equations connected with peculiar curves (1985) Iteration Theory and Its Functional Equations Lecture Notes in Mathematics, 1163. , Berlin/New York: Springer-Verlag. p. 33-44
  • Dubuc, S., Interpolation through an iterative scheme (1986) J. Math. Anal. Appl., 114, pp. 185-204
  • Dubuc, S., Elqortobi, A., Approximations of fractal sets (1990) J. Comput. Appl. Math., 29, pp. 79-89
  • Geronimo, J.S., Hardin, D.P., Massopust, P.R., Fractal functions and wavelet expansions based on several scaling functions (1994) J. Approx. Theory, 78, pp. 373-401
  • Gröchenig, K., Madych, W., Multiresolution analysis, Haar bases and selfsimilar tilings ofRn (1992) IEEE, Trans. Inform. Theory, 38, pp. 556-568
  • Handy, C., Mantica, G., Inverse problems in fractal construction: Moment method solution (1990) Physica D, 43, pp. 17-36
  • Hata, M., On the functional equation... (1985) J. Math. Kyoto Univ., 25, pp. 357-364
  • Hata, M., Fractals in mathematics (1986) Patterns and Waves - Qualitative Analysis of Nonlinear Differential Equations Stud. Math. Appl. 18, , Amsterdam-New York: North-Holland. p. 259-278
  • Heil, C., Colella, D., Matrix refinement equations: Existence and uniqueness (1996) J. Fourier Analysis Appl., 2, pp. 363-377
  • Hutchinson, J., Fractals and self-similarity (1981) Indiana Univ. Math. J., 30, pp. 713-747
  • Hwang, W.-L., Mallat, S., Characterization of self-similar multifractals with wavelet maxima (1994) Appl. Comput. Harmonic Anal., 1, pp. 316-328
  • Strichartz, R., Wavelet expansions of fractal measures (1991) J. Geom. Anal., 1, pp. 269-289
  • Strichartz, R., Wavelets and self-affine tilings (1993) Constr. Approx., 9, pp. 327-346
  • Strichartz, R., Self-similarity in harmonic analysis (1994) J. Fourier Anal. Appl., 1, pp. 1-37

Citas:

---------- APA ----------
Cabrelli, C.A. & Molter, U.M. (1999) . Generalized Self-Similarity. Journal of Mathematical Analysis and Applications, 230(1), 251-260.
http://dx.doi.org/10.1006/jmaa.1998.6200
---------- CHICAGO ----------
Cabrelli, C.A., Molter, U.M. "Generalized Self-Similarity" . Journal of Mathematical Analysis and Applications 230, no. 1 (1999) : 251-260.
http://dx.doi.org/10.1006/jmaa.1998.6200
---------- MLA ----------
Cabrelli, C.A., Molter, U.M. "Generalized Self-Similarity" . Journal of Mathematical Analysis and Applications, vol. 230, no. 1, 1999, pp. 251-260.
http://dx.doi.org/10.1006/jmaa.1998.6200
---------- VANCOUVER ----------
Cabrelli, C.A., Molter, U.M. Generalized Self-Similarity. J. Math. Anal. Appl. 1999;230(1):251-260.
http://dx.doi.org/10.1006/jmaa.1998.6200