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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A scaling equation with only
non-measurable orthogonal solutions


Author: J. Cnops
Journal: Proc. Amer. Math. Soc. 128 (2000), 1975-1979
MSC (1991): Primary 28A20, 42C15
Published electronically: November 1, 1999
MathSciNet review: 1646315
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Abstract: In this paper we construct a non-measurable scaling function for the scaling equation

\begin{displaymath}\phi(x)=\phi(3x)+\phi(3x-2)+\phi(3x-4), \end{displaymath}

using the Axiom of Choice. We prove that -apart from being non-measurable- it satisfies the classical conditions for a scaling function to lead to orthonormal wavelets. While non-measurable functions are not directly useful for numerical calculations, the example given here explains the possible anomalous behaviour of numerical methods. Indeed the origin of this paper lies in the solution of the scaling equation above, calculated on a finite grid consisting of the points $i/3^k$, $0\leq i<2.3^k$. The result is

\begin{displaymath}\phi\left(\frac i{3^k}\right)=\left\{ \begin{array}{rl} 0&i\mbox{\ odd},\\ 1&i\mbox{\ even}. \end{array}\right. \end{displaymath}

This finite approximation seemingly satisfies the conditions for a scaling equation but, as we show here, any extension is either non-measurable or not orthogonal. While in this case, and for the chosen grid, the strange behaviour is apparent from the graph of the approximating function which looks like a comb, there is of course no guarantee that this will be clear in different situations.


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Additional Information

J. Cnops
Affiliation: RUG/VWA, Galglaan 2, B-9000 Gent, Belgium
Email: jc@cage.rug.ac.be

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05188-6
PII: S 0002-9939(99)05188-6
Received by editor(s): May 14, 1998
Received by editor(s) in revised form: August 15, 1998
Published electronically: November 1, 1999
Additional Notes: The author is a post-doctoral Fellow of the FWO, Belgium.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society