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Multiscale decompositions on bounded domains

Authors: A. Cohen, W. Dahmen and R. DeVore
Journal: Trans. Amer. Math. Soc. 352 (2000), 3651-3685
MSC (2000): Primary 41A63, 42C15
Published electronically: April 17, 2000
MathSciNet review: 1458320
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Abstract | References | Similar Articles | Additional Information


A construction of multiscale decompositions relative to domains $\Omega\subset \mathbb{R} ^d$ is given. Multiscale spaces are constructed on $\Omega$ which retain the important features of univariate multiresolution analysis including local polynomial reproduction and locally supported, stable bases.

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  • [A] R. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 56:9247
  • [BL] J. Bergh and J. Löfstrom, Interpolation Spaces, an Introduction, Grundlehren der Mathematischen Wissenschaften 223, Springer, 1986. MR 58:2349
  • [BDR] C. de Boor, R. A. de Vore and A. Ron, On the construction of multivariate (pre) wavelets, Constructive Approximation 9 (1993), 123-166. MR 94k:41048
  • [CDP] J. M. Carnicer, W. Dahmen and J. M. Peña, Local decomposition of refinable spaces, in preparation.
  • [CDM] A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary Subdivision, Mem. Amer. Math. Soc., Vol. 93, No. 453 (1991). MR 92h:65017
  • [CQ] C. K. Chui and E. Quak, Wavelets on a bounded interval, in Numerical Methods of Approximation Theory, Vol. 9, D. Braess and L. L. Schumaker, eds., International Series of Numerical Mathematics, Vol. 105, Birkhäuser, Basel, 1992. MR 95b:42027
  • [CD] A. Cohen and I. Daubechies, Nonseparable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9 (1993), 51-137. MR 94k:42047
  • [CDF] A. Cohen, I. Daubechies and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), 485-560. MR 93e:42044
  • [CDV] A. Cohen, I. Daubechies and P. Vial, Wavelets and fast wavelet transforms on the interval, ACHA 1 (1994), 54-81. MR 94m:42074
  • [D] W. Dahmen, Some remarks on multiscale transformations, stability and biorthogonality, in Curves and surfaces, P. J. Laurent, A. Le Méhauté, L. L. Schumaker, eds., Academic Press, 1994. MR 92c:65006
  • [D2] W. Dahmen, Stability of multiscale transformations, preprint, Fourier Anal. Appl. 2 (1996), 341-361. MR 97ki:46133
  • [DDS] W. Dahmen, R. A. DeVore and K. Scherer, Multidimensional spline approximation, SIAM J. Numer. Anal. 17 (1980), 380-402. MR 81j:41015
  • [DK] W. Dahmen and A. Kunoth, Multilevel preconditioning, Numer. Math. 63 (1992), 315-344. MR 93j:65065
  • [DM93] W. Dahmen, C. A. Micchelli, Biorthogonal wavelet expansions, Constructive Approximation 13 (1997), 293-328. MR 99c:39039
  • [Dau] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996. MR 90m:42039
  • [Dau2] I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Math., 61, SIAM, 1992. MR 93e:42045
  • [DJP] R. DeVore, B. Jawerth and V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992), 737-785. MR 94a:42025
  • [DL] R. DeVore and G. Lorentz, Constructive Approximation, Grundlehren der Mathematischen Wissenschaften, Springer, 1993. MR 95f:41001
  • [DP] R. DeVore and V. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305 (1988), 397-414. MR 89h:46044
  • [DS] R. DeVore and R. Sharpley, Maximal functions measuring smoothness, Mem. Amer. Math. Soc., No. 293, 1983. MR 85g:46039
  • [JM] R. Q. Jia and C. A. Micchelli, Using the refinement equation for the construction of pre-wavelets II: Powers of two, in Curves and Surfaces, P. J. Laurent, A. LeMéhauté, L. L. Schumaker, eds., Academic Press, New York, 1991, pp. 209-246. MR 93e:65024
  • [JS] H. Johnen and K. Scherer, On the equivalence of the $K$-functional and moduli of continuity and some applications, in Constructive Theory of Functions of Several Variables, Lecture Notes in Math., Vol. 571, Springer, 1977, pp. 119-140. MR 58:7060
  • [LM] P. G. Lemarié and G. Malgouyres, Support des fonctions de base dans une analyse multiresolution, C. R. Acad. Sci. Paris 213 (1991), 377-380. MR 92k:42044
  • [M] V. G. Maz'ja, Sobolev Spaces, Springer-Verlag, Berlin, 1985. MR 87g:46056
  • [Me] Y. Meyer, Ondelettes et Opérateurs, Vols. 1 and 2, Hermann, Paris, 1990. MR 93i:42002; MR 93i:42003
  • [OS] P. Oswald and E. A. Storozhenko, Jackson's theorem in the spaces $L_p(\mathbb{R} ^k)$, $0<p<1$, Siberian. Math. 19 (1978), 630-639. MR 58:12154
  • [Sch] I. J. Schoenberg, Contribution to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math. 4 (1946), 45-99, 112-141. MR 7:487b; MR 8:55d
  • [S] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970.
  • [T] H. Triebel, Interpolation Theory Function Spaces and Differential Operators, North-Holland, Amsterdam, 1978. MR 80i:46032b

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Additional Information

A. Cohen
Affiliation: Laboratoire d’Analyse Numerique, Université de Paris VI, 4 Place Jussieu, 75005 Paris, France

W. Dahmen
Affiliation: Rheinisch-Westf Technische Hochscule, Templergraben 55, D-52052 Aachen, Germany

R. DeVore
Affiliation: Industrial Mathematics Institute, University of South Carolina, Columbia, South Carolina 29208-0001

Received by editor(s): December 28, 1995
Received by editor(s) in revised form: February 26, 1997
Published electronically: April 17, 2000
Additional Notes: This research was supported by ONR Contract N0014-91-J1343 and a NATO travel grant
Article copyright: © Copyright 2000 American Mathematical Society

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