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Fractional Wavelets, Derivatives, and Besov Spaces

M. Unser, T. Blu

Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing X, San Diego CA, USA, August 4-8, 2003, vol. 5207, part I, pp. 147-152.


We show that a multi-dimensional scaling function of order γ (possibly fractional) can always be represented as the convolution of a polyharmonic B-spline of order γ and a distribution with a bounded Fourier transform which has neither order nor smoothness. The presence of the B-spline convolution factor explains all key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multi-scale differentiation property, and smoothness of the basis functions. The B-spline factorization also gives new insights on the stability of wavelet bases with respect to differentiation. Specifically, we show that there is a direct correspondence between the process of moving a B-spline factor from one side to another in a pair of biorthogonal scaling functions and the exchange of fractional integrals/derivatives on their wavelet counterparts. This result yields two “eigen-relations” for fractional differential operators that map biorthogonal wavelet bases into other stable wavelet bases. This formulation provides a better understanding as to why the Sobolev/Besov norm of a signal can be measured from the lp-norm of its rescaled wavelet coefficients. Indeed, the key condition for a wavelet basis to be an unconditional basis of the Besov space Bqs(Lp(Rd)) is that the s-order derivative of the wavelet be in Lp.

@INPROCEEDINGS(http://bigwww.epfl.ch/publications/unser0307.html,
AUTHOR="Unser, M. and Blu, T.",
TITLE="Fractional Wavelets, Derivatives, and {B}esov Spaces",
BOOKTITLE="Proceedings of the {SPIE} Conference on Mathematical
	Imaging: {W}avelet Applications in Signal and Image Processing {X}",
YEAR="2003",
editor="",
volume="5207",
series="",
pages="147--152",
address="San Diego CA, USA",
month="August 3-8,",
organization="",
publisher="",
note="Part {I}")

© 2003 SPIE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from SPIE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.
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