RUI: Orthonormal Fourier Bases and Iterated Function Systems

Grinnell College, Grinnell IA

Investigators

Abstract

ABSTRACT The investigators study Fourier bases of Hilbert spaces associated with affine iterated function systems (IFSs) on the real line and in higher dimensions. While much work in the last two decades has focused on the case in which the attractor of the IFS is a fractal with non-integral Hausdorff dimension, the investigators study IFSs with overlap. In the one-dimensional setting, the measures associated with these attractors are Erdos's self-similar convolution measures; in general the measures are invariant Hutchinson measures. Almost nothing is known about orthonormal bases associated with Hutchinson measures in the case of non-finite overlap, and in fact previous work of the investigators and P. Jorgensen has shown that the measures in the overlap cases have different properties from the non-overlap measures. The proposed work on IFSs is an exciting branch of harmonic analysis which connects to fractals, wavelets, and random walks; it also has applications to number theory, dynamics, and combinatorial geometry. The investigators draw on methods from all these fields in their work. The work in this proposal will involve further collaborations with P. Jorgensen at the University of Iowa. Iterated function systems (IFSs) embody the common themes of recursion and self-similarity. The investigators will study properties of IFSs and their associated geometry. In fact, a central theme in this work is the interplay between geometry and spectral analysis on fractals. There is a natural tension between studying fractals from a geometric point of view and from a spectral point of view; traditional time-frequency methods used in spectral analysis are linear, and the systems the investigators study are non-linear. The need to understand non-linear phenomena is motivated by a host of real-life applications. IFSs occur repeatedly in signal processing and communications. For example, wavelets, the key behind the JPEG 2000 compression standard, are a particular example of an IFS. Applications of iterated function systems extend far beyond communications---IFSs are used in data compression, quantum computing, pattern recognition, DNA computations, and a vast array of other fields. Furthermore, due to its wide applicability, work in this proposal will draw undergraduate students into mathematical research. The investigators will conduct research with undergraduates at Grinnell College, and published work by P. Jorgensen and the investigators has already been incorporated into research projects for undergraduates at the University of Iowa, as part of the Alliances for Graduate Education and the Professoriate (AGEP Alliance) and the Heartland Mathematics Partnership.