ENVIRONMENTAL POLICY CONSIDERATIONS IN THE GRAIN-LIVESTOCK SUBSECTORS IN CANADA, MEXICO AND THE UNITED STATES

Environmental Economics and Policy,

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 $r: z \mapsto z^2$ in the one-torus \bt, a wavelet filter, and an associated transfer operator. This leads to a scaling function and a corresponding closed subspace $V_0$ in the Hilbert space L^2(\br). Using the dyadic scaling on the line \br, one has a nested family of closed subspaces $V_n$, 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 $r$ (i.e., generalizing $r(z) = z^2$) in a compact metric space $X$. It is assumed that $r : X\to X$ 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 $H^2$. By this we mean that there are lacunary subsets $\Gamma$ of the non-negative integers, and associated closed $\Gamma$-subspace in the Hardy space H^2(\bd), \bd denoting the disk, such that for every function $f$ in in $H^2(\Gamma)$, and for every point $z$ in \bd, $f(z)$ admits a boundary integral represented by an associated measure $\mu$, with integration over \supp{\mu} placed as a Cantor subset on the circle \bt := \{bd}(\bd). We study families of pairs: measures $\mu$ and sets $\Gamma$ of lacunary form, admitting lacunary Fourier series in $L^2(\mu)$; i.e., configurations $\Gamma$ arranged with a geometric progression of empty spacing, or missing parts, gaps. Given $\Gamma$, we find corresponding generalized Szeg\" o kernels $G_\Gamma$, and we compare them to the classical Szeg\" o kernel for \bd. Rather than the more traditional approach of starting with $\mu$, and then asking for possibilities for sets $\Gamma$, such that we get Fourier series representations, we turn the problem upside down; now starting instead with a countably infinite discrete subset $\Gamma$, and, within a new duality framework, we study the possibilities for choices of measures $\mu$

Isospectral measures

In recent papers a number of authors have considered Borel probability measures $\mu$ in \br^d such that the Hilbert space $L^2(\mu)$ has a Fourier basis (orthogonal) of complex exponentials. If $\mu$ satisfies this property, the set of frequencies in this set are called a spectrum for $\mu$. Here we fix a spectrum, say $\Gamma$, and we study the possibilities for measures $\mu$ having $\Gamma$ as spectrum.Comment: v