Karen L. Shuman

Orthogonal Exponentials for Bernoulli Iterated Function Systems

We investigate certain spectral properties of the Bernoulli convolution measures on attractor sets arising from iterated function systems (IFSs) on ℝ. In particular, we examine collections of orthogonal exponential functions in the Hilbert space of square-integrable functions on the attractor. We carefully examine a test case λ = 3/4 in which the IFS has overlap. We also determine rational λ = a/b for which infinite sets of orthogonal exponentials exist.

HARMONIC ANALYSIS OF ITERATED FUNCTION SYSTEMS WITH OVERLAP

An iterated function system (IFS) is a system of contractive mappings τi:Y→Y, i=1,…,N (finite), where Y is a complete metric space. Every such IFS has a unique (up to scale) equilibrium measure (also called the Hutchinson measure μ), and we study the Hilbert space L2(μ). In this paper we extend previous work on IFSs without overlap. Our method involves systems of operators generalizing the more familiar Cuntz relations from operator algebra theory and from subband filter operators in signal processing. These Cuntz-like operator systems were used in recent papers on wavelet analysis by Baggett, Jorgensen, Merrill, and Packer [Contemp. Math. 345, 11–25 (2004)], where they serve as a first step to generating wavelet bases of Parseval type (alias normalized tight frames), i.e., wavelet bases with redundancy. Similarly, it was shown in work by Dutkay and Jorgensen [Rev. Mat. Iberoam. 22, 131–180 (2006)] that the iterative operator approach works well for generating wavelets on fractals from IFSs without overla...

An Operator-Fractal

Certain Bernoulli convolution measures μ are known to be spectral. Recently, much work has concentrated on determining conditions under which orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure known to be spectral, the ONB need not be unique; indeed, there are often families of such spectral bases. Let for a natural number n and consider the Bernoulli measure with scale factor λ. It is known that L 2(μλ) has a Fourier basis. We first show that there are Cuntz operators acting on this Hilbert space which create an orthogonal decomposition, thereby offering powerful algorithms for computations for Fourier expansions. When L 2(μλ) has more than one Fourier basis, there are natural unitary operators U, indexed by a subset of odd scaling factors p; each U is defined by mapping one ONB to another. We show that the unitary operator U can also be orthogonally decomposed according to the Cuntz relations. Moreover, this operator-fractal U exhibits its own self-similarity.

Scalar spectral measures associated with an operator-fractal

We examine the operator

Complete signal processing bases and the Jacobi group

U_5

Tests of a new basis for signal processing

defined on

Affine Systems: Asymptotics at Infinity for Fractal Measures

L^2(\mu_{\frac14})

Families of Spectral Sets for Bernoulli Convolutions

where

Iterated Function Systems, Moments, and Transformations of Infinite Matrices

\mu_{\frac14}

Weak Convergence of Greedy Algorithms in Banach Spaces

is the 1/4 Cantor measure. The operator