*Journal of Mathematical Analysis and Applications* features research by Dr. Michael Wilson

**Release Date:** 04-08-2009

**Author:** Dawn Densmore

**Email:** Dawn.Densmore@uvm.edu

A
research paper authored by Dr. Michael Wilson, professor in the Department
of Mathematics and Statistics within the UVM College of Engineering and
Mathematical Sciences (CEMS), has been published in the May 2009
*Journal of Mathematical Analysis and Applications*. The paper,
"Stability of wavelet-like expansions under chromatic aberration," proves
that wavelet and wavelet-like expansions of functions are Lp-stable under
small errors in translation and dilation of the constituent-reproducing
kernels.

"Wavelets and their close relatives provide a way of analyzing signals that is stable with respect to small errors in the data," says Dr. Wilson. "Fourier series and Fourier transforms donâ€™t do that. If nature hands you a function, the only way you can understand it is by sampling, and that introduces errors into everything you will derive about the function." According to Wilson, the effect of these errors on the Fourier transform, even if very small, can be disastrous whereas wavelets are more forgiving. His paper shows that wavelets stay forgiving even if the data is corrupted in ways that correspond to chromatic aberration by introducing arbitrary but small errors in the processing of a signal, which are allowed to vary with frequency and position.

The *Journal of Mathematical Analysis and Applications* presents
papers that treat mathematical analysis and its numerous applications and
emphasizes articles devoted to the mathematical treatment of questions
arising in physics, chemistry, biology, and engineering, particularly those
that stress analytical aspects and novel problems and their solutions.