Asymptotic error expansion of
wavelet approximations of smooth functions II |
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Authors: | Wim Sweldens Robert Piessens |
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Institution: | (1) University of South Carolina, Department of Mathematics, Columbia, SC 29208, USA , US;(2) Katholieke Universiteit Leuven, Department of Computer Science, Celestijnenlaan 200A, B-3001 Leuven, Belgium e-mail: sweldens@math.scarolina.edu , BE |
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Abstract: | Summary. We generalize earlier results concerning
an asymptotic error expansion of wavelet
approximations. The properties of the monowavelets,
which are the building
blocks for the error expansion, are studied in more
detail, and connections
between spline wavelets and Euler and
Bernoulli polynomials are pointed out.
The expansion is used to compare the
error for different wavelet families.
We prove that the leading terms of the
expansion only depend on the multiresolution
subspaces and not
on how the complementary subspaces
are chosen.
Consequently, for a fixed set of
subspaces , the leading
terms do not depend on the fact whether
the wavelets are orthogonal or not.
We also show that Daubechies' orthogonal wavelets need,
in general, one level more than spline wavelets to obtain an
approximation with a prescribed accuracy.
These results are illustrated with numerical examples.
Received May 3, 1993 / Revised version received January 31, 1994 |
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Keywords: | Mathematics Subject Classification (1991): 41A30 65D32 42C05 |
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