1,544 research outputs found
Hilbert spaces built on a similarity and on dynamical renormalization
We develop a Hilbert space framework for a number of general multi-scale
problems from dynamics. The aim is to identify a spectral theory for a class of
systems based on iterations of a non-invertible endomorphism.
We are motivated by the more familiar approach to wavelet theory which starts
with the two-to-one endomorphism in the one-torus \bt, a
wavelet filter, and an associated transfer operator. This leads to a scaling
function and a corresponding closed subspace in the Hilbert space
L^2(\br). Using the dyadic scaling on the line \br, one has a nested family
of closed subspaces , n \in \bz, with trivial intersection, and with
dense union in L^2(\br). More generally, we achieve the same outcome, but in
different Hilbert spaces, for a class of non-linear problems. In fact, we see
that the geometry of scales of subspaces in Hilbert space is ubiquitous in the
analysis of multiscale problems, e.g., martingales, complex iteration dynamical
systems, graph-iterated function systems of affine type, and subshifts in
symbolic dynamics. We develop a general framework for these examples which
starts with a fixed endomorphism (i.e., generalizing ) in a
compact metric space . It is assumed that is onto, and
finite-to-one.Comment: v3, minor addition
Affine fractals as boundaries and their harmonic analysis
We introduce the notion of boundary representation for fractal Fourier
expansions, starting with a familiar notion of spectral pairs for affine
fractal measures. Specializing to one dimension, we establish boundary
representations for these fractals. We prove that as sets these fractals arise
as boundaries of functions in closed subspaces of the Hardy space . By
this we mean that there are lacunary subsets of the non-negative
integers, and associated closed -subspace in the Hardy space
H^2(\bd), \bd denoting the disk, such that for every function in in
, and for every point in \bd, admits a boundary
integral represented by an associated measure , with integration over
\supp{\mu} placed as a Cantor subset on the circle \bt := \{bd}(\bd).
We study families of pairs: measures and sets of lacunary
form, admitting lacunary Fourier series in ; i.e., configurations
arranged with a geometric progression of empty spacing, or missing
parts, gaps. Given , we find corresponding generalized Szeg\" o kernels
, and we compare them to the classical Szeg\" o kernel for \bd.
Rather than the more traditional approach of starting with , and then
asking for possibilities for sets , such that we get Fourier series
representations, we turn the problem upside down; now starting instead with a
countably infinite discrete subset , and, within a new duality
framework, we study the possibilities for choices of measures
Isospectral measures
In recent papers a number of authors have considered Borel probability
measures in \br^d such that the Hilbert space has a Fourier
basis (orthogonal) of complex exponentials. If satisfies this property,
the set of frequencies in this set are called a spectrum for . Here we fix
a spectrum, say , and we study the possibilities for measures
having as spectrum.Comment: v
On the universal tiling conjecture in dimension one
We show that the spectral-tile implication in the Fuglede conjecture in
dimension 1 is equivalent to a Universal Tiling Conjecture and also to similar
forms of the same implication for some simpler sets, such as unions of
intervals with rational or integer endpoints.Comment: arXiv admin note: text overlap with arXiv:1205.435
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