ANALYSIS  SEMINAR (UTK)

Abstract:. We will explain how one may "weld" two Blaschke products to obtain a rational function, and address the question of whether the rational function one obtains by this procedure is unique. We will also explain how this question arose in the recent literature on applications of complex analysis to pattern recognition. Along the way we will talk about some relevant work of UTK professors Ken Stephenson and Carl Sundberg.   

Abstract: If $\mathcal H$ denotes a Hilbert space of analytic functions in the open unit disc $\mathbb D\subseteq \mathbb C$, then a function $f\in \mathcal H$ is called cyclic, if the polynomial multiples of $f$ are dense in $\mathcal H$. Cyclic functions cannot have any zeros in the open unit disc, and for many spaces $\mathcal H$ it is interesting to characterize cyclicity in terms of how small $|f(z)|$ can be near the boundary of $\mathbb D$. For example, if $\mathcal H=H^2$, the space of analytic functions in $\mathbb D$ whose Taylor coefficients are square summable, then $f$ is cyclic in $H^2$, if and only if it is a so-called outer function, i.e.  the harmonic function $\log|f(z)|$ equals the Poisson integral of an absolutely continuous measure. This excludes for example functions of the type $f(z)=e^{-\frac{1+z}{1-z}}$ where $|f|$ has exponential decline on the radius $\to 1$, and $\log |f(z)|$ equals the Poisson integral of a point mass. 

Let $D=\{f\in \mathrm{Hol}(\mathbb D): \int_{\mathbb D} |f'|^2dxdy<\infty\} $ denote the Dirichlet space of the unit disc $\mathbb D$. One checks that $D\subseteq H^2$ with $\|f\|_{H^2}\le \|f\|_D$. It follows that cyclic functions in $D$ must be outer, but deciding which outer functions  are cyclic in $D$ is an open problem. We show that an outer function  $f\in D$ is cyclic in $D$, whenever $\log f$ belongs to the Pick-Smirnov class $N^+(D)$. If $f$ is a bounded analytic function in $D$, then cyclicity can also be checked via iterated logarithms. For example, we show that an outer function $f$ is cyclic, whenever $\log(1+ \log(1/f))\in N^+(D)$. This condition can be checked by verifying that $\log(1+ \log(1/f))\in D$. If $f$ satisfies a mild extra condition, then the conditions also become necessary for cyclicity. The talk is based on joint work with Alexandru Aleman.

Abstract:. The study of the boundary behavior of quasiconformal mappings started with Beurling and Ahlfors. They proved that any quasiconformal mapping of the upper half plane extends to a quasisymmetric mapping of the real line and vice versa. Since then, quasisymmetric mappings have played a central role in Geometric Function Theory and have been used to characterize domains and mappings. In this talk, we will look at a finer version of such embeddings and use them to characterize certain domains in the plane. 

Abstract:. A classical uniformization theorem of Ahlfors from 1963 asserts that every bounded turning planar arc is the image of the unit segment under a quasiconformal self-map of the plane. We prove an analogous uniformization theorem for planar trees by showing that every planar quasiconformal tree with uniform branch separation is quasiconformally equivalent to a quasiconvex planar tree with uniform branch separation and Hausdorff dimension arbitrarily close to 1. Along the way, we show that every quasisymmetric (resp. bi-Lipschitz) homeomorphism between planar quasiconformal trees that respects the counterclockwise orientation of the branches at every branch point can be extended quasisymmetrically (resp. bi-Lipscitz) to the whole plane. As a corollary of the extension theorem we construct a large class of quadrivalent quasiconformal trees, which includes the Vicsek fractal, so that each tree in this class is the Julia set of a uniformly quasiregular map of the plane.