14 research outputs found
Families of spectral sets for Bernoulli convolutions
In this paper, we study the harmonic analysis of Bernoulli measures. We show
a variety of orthonormal Fourier bases for the L^2 Hilbert spaces corresponding
to certain Bernoulli measures, making use of contractive transfer operators.
For other cases, we exhibit maximal Fourier families that are not orthonormal
bases.Comment: 25 pages, same result
Additive spectra of the 1/4 Cantor measure
In this paper, we add to the characterization of the Fourier spectra for
Bernoulli convolution measures. These measures are supported on Cantor subsets
of the line. We prove that performing an odd additive translation to half the
canonical spectrum for the 1/4 Cantor measure always yields an alternate
spectrum. We call this set an additive spectrum. The proof works by connecting
the additive set to a spectrum formed by odd multiplicative scaling.Comment: 9 pages, 1 figur
Scaling by 5 on a 1/4-Cantor Measure
Each Cantor measure (\mu) with scaling factor 1/(2n) has at least one
associated orthonormal basis of exponential functions (ONB) for L^2(\mu). In
the particular case where the scaling constant for the Cantor measure is 1/4
and two specific ONBs are selected for L^2(\mu), there is a unitary operator U
defined by mapping one ONB to the other. This paper focuses on the case in
which one ONB (\Gamma) is the original Jorgensen-Pedersen ONB for the Cantor
measure (\mu) and the other ONB is is 5\Gamma. The main theorem of the paper
states that the corresponding operator U is ergodic in the sense that only the
constant functions are fixed by U.Comment: 34 page
Harmonic analysis of iterated function systems with overlap
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.Comment: 37 page
Convolutional Frames and the Frame Potential
The recently introduced notion of frame potential has proven useful for the characterization of finite-dimensional tight frames. The present work represents an effort to similarly characterize finite-dimensional tight frames with additional imposed structure. In particular, it is shown that the frame potential still leads to a complete description of tight frames when restricted to the class of translation-invariant systems. It is natural to refer to such frames as convolutional because of the correspondence between translation-invariant systems and finite-dimensional filter banks. The fast algorithms associated with convolution represent one possible advantage over nonconvolutional frames in applications.Abstract © Elsevie