IJPAM: Volume 31, No. 2 (2006)

THE SPHERICAL BERNSTEIN WAVELET

M.J. Fengler$^1$, W. Freeden$^2$, M. Gutting$^3$
$^{1,2,3}$Geomathematics Group
Department of Mathematics
University of Kaiserslautern
P.O. Box 3049, Kaiserslautern, 67653, GERMANY
$^1$e-mail: fengler@mathematik.uni-kl.de
$^2$e-mail: freeden@mathematik.uni-kl.de
$^3$e-mail: gutting@mathematik.uni-kl.de


Abstract.In this work we introduce a new bandlimited spherical wavelet: The Bernstein wavelet. It possesses a couple of interesting properties. To be specific, we are able to construct bandlimited wavelets free of oscillations. The scaling function of this wavelet is investigated with regard to the spherical uncertainty principle, i.e., its localization in the space domain as well as in the momentum domain is calculated and compared to the well-known Shannon scaling function. Surprisingly, they possess the same localization in space although one is highly oscillating whereas the other one shows no oscillatory behavior. Moreover, the Bernstein scaling function turns out to be the first bandlimited scaling function known to the literature whose uncertainty product tends to the minimal value $1$.

Received: July 17, 2006

AMS Subject Classification: 33C55, 42C10, 42C15, 42C40, 65T99

Key Words and Phrases: spherical wavelets, Bernstein polynomials, constructive approximation on the sphere, uncertainty principle

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 31
Issue: 2