Abstract
A unified abstract framework for the multilevel decomposition of both Banach and quasi-Banach spaces is presented. The characterization of intermediate spaces and their duals is derived from general Bernstein and Jackson inequalities. Applications to compactly supported biorthogonal wavelet decompositions of families of Besov spaces are also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bergh, J. and Löfström, J. (1976)Interpolation spaces, an introduction, Springer Verlag, Berlin.
Canuto, C. and Tabacco, A. (1998)Ondine biortogonali: Teoria e Applicazioni, Quaderno UMI, in preparation.
Canuto, C. and Tabacco, A.Absolute and relative cut-off in adaptive approximation by wavelets, in preparation.
Cohen, A., Daubechies, I. and Feauveau, J. (1992) Biorthogonal bases of compactly supported wavelet,Comm. Pure and Appl. Math. 45, 485–560.
Dahlke, S. and DeVore, R.A. (1995)Besov regularity for elliptic boundary value problems, preprint.
Dahmen, W. (1993) Decomposition of refinable spaces and applications to operator equations,Algorithms for Approximation III, M.G. Cox, J.C. Mason, Eds., Baltzer, Amsterdam.
Dahmen, W. (1995) Multiscale analysis, approximation and interpolation spaces,Approximation Theory VIII, 1–41, C.K. Chui and L.L. Schumaker, Eds., World Scientific Pub., Singapore.
Dahmen, W. (1996)Stability of multiscale transformations, J. Fourier Anal. and Appl. 4, 341–362.
Dahmen, W. and Kunoth, A. (1992) Multilevel Preconditioning,Numer. Math. 63, 315–344.
Daubechies, I. (1992)Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics 61, SIAM, Philadelphia.
DeVore, R.A., Jawerth, B. and Lucier, B.J. (1992) Image compression through wavelet transform coding,IEEE Trans. Information Th.,38, 719–746.
DeVore, R.A., Jawerth, B. and Popov, V.A. (1992) Compression of wavelet decompositions,Amer. J. Math. 114, 737–785.
DeVore, R.A. and Lorentz, G.G. (1993)Constructive Approximation, Springer Verlag, Berlin.
DeVore, R.A. and Lucier, B.J. (1992) Fast wavelet techniques for near-optimal image processing,Proc. IEEE Mil. Comm. Conf., IEEE Comm. Soc., New York.
DeVore, R.A. and Popov, V.A. (1998) Interpolation spaces and non-linear approximation,Function Spaces and Applications, 191–205, M. Cwikel, J. Peetre, Y. Sagher and H. Wallin, Eds., Lecture Notes in Math. 1302, Springer Verlag, Berlin, 1988.
DeVore, R.A. and Popov, V.A. (1998) Interpolation of Besov spaces,Trans. Amer. Soc. 305, 397–414.
Frazier, M. and Jawerth, B. (1985) Decomposition of Besov spaces,Indiana U. Math. J. 34, 777–799.
Lemarie-Rieusset, P.-G. and Malgouyres, G. (1991) Support des fonctions de base dans une analyse multirésolution,C. R. Acad. Sci. Paris,33, 377–380.
Kunoth, A. (1994)Multilevel Preconditioning, PhD. Thesis, Verlag-Shaker, Aachen.
Mallat, S. (1989) Multiresolution approximation and wavelet orthonormal bases ofl 2,Trans Amer. Math. Soc. 315, 69–88.
Meyer, Y. (1990)Ondelettes, vol.I: Ondelettes et Opèrateurs, Hermann, Paris.
Peetre, J. and Sparr, G. (1972) Interpolation of normed Abelian groups,Ann. Mat. Pura Appl. 92, 217–262.
Author information
Authors and Affiliations
Additional information
The first author was partially supported by grants from MURST (40% Analisi Numerica) and ASI (Contract ASI-92-RS-89), whereas the second author was partially supported by grants from MURST (40% Analisi Funzionale) and CNR (Progetto Strategico “Applicazioni della Matematica per la Tecnologia e la Società”).
Rights and permissions
About this article
Cite this article
Canuto, C., Tabacco, A. Multilevel decompositions of functional spaces. The Journal of Fourier Analysis and Applications 3, 715–742 (1997). https://doi.org/10.1007/BF02648264
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02648264