[en] We present a construction of regular compactly supported wavelets in any Sobolev space of integer order. It is based on the existence and suitable estimates of filters defined from polynomial equations. We give an implicit study of these filters and use the results obtained to construct scaling functions leading to multiresolution analysis and wavelets. Their regularity increases linearly with the length of their supports as in the L(2) case. One technical problem is to prove that the intersection of the scaling spaces is reduced to 0. This is solved using sharp estimates of Littlewood-Paley type. (C) 1997 Academic Press, Inc.
Disciplines :
Physics Mathematics
Author, co-author :
Bastin, Françoise ; Université de Liège - ULiège > Département de mathématique > Analyse, analyse fonctionnelle, ondelettes
Laubin, P.
Language :
English
Title :
Compactly supported wavelets in Sobolev spaces of integer order
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Bibliography
F. Bastin and P. Laubin, Regular compactly supported wavelets in Sobolev spaces, preprint.
I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41(7), 1988.
I. Daubechies, "Ten Lectures on Wavelets," CBMS, Vol. 61, SIAM, Philadelphia, 1992.
Y. Meyer, "Ondelettes et opérateurs, I," Hermann, Paris, 1990.
S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(ℝ), Trans. Amer. Math. Soc. 315(1), 1989.
W. Sweldens, "The Construction and Application of Wavelets in Numerical Analysis," Doctoral Thesis, KUL, 1994.
G. Walter, "Wavelets and Other Orthogonal Systems with Applications," CRC Press, Boca Raton, FL, 1994.
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