Multiscale Operators and Systems of Biorthogonal Functions
by
S. L. Lee
Department of Mathematics, National University of Singapore

In this talk we describe new results on the eigenvalues of scaling operators and their adjoints and construct the corresponding biorthogonal systems of eigenfuctions. The biorthogonal systems that arise from a scaling operator consist of the distributional derivatives of the corresponding scaling function and a sequence of polynomials. For the case in which the scaling function is a B-spline, the corresponding biorthogonal polynomials are Bernoulli polynomials of the same order as the B-spline. In particular, for the simplest case of Haar function the polynomials that are biorthogonal to the distributional derivatives of the Haar function are the Bernoulli polynomials. On the other hand, for the Gaussian, which is a scaling function, the corresponding biorthogonal polynomials are the Hermite polynomials.

Date received: February 6, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caie-18.