arXiv subjects

Trevor Richards

Publications and source records attributed to Trevor Richards.

10 recordsLinked to original sources

arXiv2019

Boundary convergence and path divergence sets for bounded analytic functions in the disk

Let $f:\mathbb{D}\to\mathbb{C}$ be a bounded analytic function. A set $K\subset\mathbb{D}$ which contains the point $1$ in its boundary is called a convergence set for $f$ at $1$ if $f(z)$ converges to some value $ζ$ as $z\to1$ with $z\in K$. $K$ is called a path divergence set for $f$ at $1$ if $f$ diverges along every path $γ$ which lies in $K$ and approaches $1$. In this article, we show that for a path $γ$ through the unit disk from $-1$ to $1$, if $f$ fails to converge along $γ$, then either the region above $γ$ or the region below $γ$ is a path divergence set for $f$. On the other hand, if $γ_1$ and $γ_2$ are two such paths, and $f$ converges along both $γ_1$ and $γ_2$, then the region between $γ_1$ and $γ_2$ is a convergence set for $f$. This latter fact is immediate when $γ_1$ and $γ_2$ do not intersect except at their end-points, but becomes non-trivial when $γ_1$ and $γ_2$ are highly intersecting. We conclude the paper with an examination of the convergence sets for the function $e^{\frac{z+1}{z-1}}$ at $1$.

math.CV↗

arXiv2018

Computing polynomial conformal models for low-degree Blaschke products

For any finite Blaschke product $B$, there is an injective analytic map $φ:\mathbb{D}\to\mathbb{C}$ and a polynomial $p$ of the same degree as $B$ such that $B=p\circφ$ on $\mathbb{D}$. Several proofs of this result have been given over the past several years, using fundamentally different methods. However, even for low-degree Blaschke products, no method has hitherto been developed to explicitly compute the polynomial $p$ or the associated conformal map $φ$. In this paper, we show how these functions may be computed for a Blaschke product of degree at most three, as well as for Blaschke products of arbitrary degree whose zeros are equally spaced on a circle centered at the origin.

math.CV↗

arXiv2017

On approximate Gauss-Lucas theorems

The Gauss--Lucas theorem states that any convex set $K\subset\mathbb{C}$ which contains all $n$ zeros of a degree $n$ polynomial $p\in\mathbb{C}[z]$ must also contain all $n-1$ critical points of $p$. In this paper we explore the following question: for which choices of positive integers $n$ and $k$, and positive real number $ε$, will it follow that for every degree $n$ polynomial $p$ with at least $k$ zeros lying in $K$, $p$ will have at least $k-1$ critical points lying in the $ε$-neighborhood of $K$. We supply an inequality relating $n$, $k$, and $ε$ which, when satisfied, guarantees a positive answer to the above question.

math.CV↗

arXiv2017

Recognizing difference quotients of real functions

For a real function $f:[0,1]\to\mathbb{R}$, the difference quotient of $f$ is the function of two real variables $\operatorname{DQ}_f(a,b)=\dfrac{f(b)-f(a)}{b-a}$, which we view as defined on the triangle $\mathcal{T}=\{(a,b):0\leq a<b\leq1\}$. In this paper we investigate how to determine whether a given function of two variables $H(a,b)$ is the difference quotient of some real function $f(x)$. We develop three independent methods for recognizing such a function $H$ as a difference quotient, and corresponding methods for recovering the underlying function $f$ from $H$.

math.CA↗

arXiv2016

Characterizing meromorphic pseudo-lemniscates

Let $f$ be a meromorphic function with simply connected domain $G\subset\mathbb{C}$, and let $Γ\subset\mathbb{C}$ be a smooth Jordan curve. We call a component of $f^{-1}(Γ)$ in $G$ a $Γ$-$pseudo$-$lemniscate$ of $f$. In this note we give criteria for a smooth Jordan curve $\mathcal{S}$ in $G$ (with bounded face $D$) to be a psuedo-lemniscate of $f$ in terms of the number of preimages (counted with multiplicity) which a given $w$ has under $f$ in $D$, as $w$ ranges over the Riemann sphere. We also develop a test, in the same terms, by which one may show that the image of a Jordan curve under $f$ is not a Jordan curve.

math.CV↗

arXiv2016

Conformal models and fingerprints of pseudo-lemniscates

We prove that every function that is meromorphic on the closure of an analytic Jordan domain and sufficiently well-behaved on the boundary is conformally equivalent to a rational map whose degree is smallest possible. We also show that the minimality of the degree fails in general without the boundary assumptions. As an application, we generalize a theorem of Ebenfelt, Khavinson and Shapiro by characterizing fingerprints of polynomial pseudo-lemniscates.

math.CV↗

arXiv2015

On Scottish Book Problem 157

This paper describes our hunt for the solver of Problem 157 in the Scottish Book, a problem originally posed by A.~J. (Gus) Ward in 1937. We first make the observation that a theorem of Richard O'Malley from 1975 yields an immediate positive solution. A further look at O'Malley's references revealed a 1970 paper by Donald Ornstein that we now believe contains the first solution of {\em SB 157}. We isolate the common elements in the machinery used by both Ornstein and O'Malley and discuss several consequences. We also examine an example function given by Ornstein. There are some difficulties with this function but we provide a fix, and show moreover that functions of that kind are typical in the sense of the Baire category theorem.

math.CA↗

arXiv2015

Conformal equivalence of analytic functions on compact sets

In this paper we present a geometric proof of the following fact. Let $D$ be a Jordan domain in $\mathbb{C}$, and let $f$ be analytic on $cl(D)$. Then there is an injective analytic map $ϕ:D\to\mathbb{C}$, and a polynomial $p$, such that $f\equiv p\circϕ$ on $D$ (that is, $f$ has a polynomial conformal model $p$).

math.CV↗

arXiv2015

Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

Let $f=B_1/B_2$ be a ratio of finite Blaschke products having no critical points on $\partial\mathbb{D}$. Then $f$ has finitely many critical level curves (level curves containing critical points of $f$) in the disk, and the non-critical level curves interpolate smoothly between the critical level curves. Thus, to understand the geometry of all the level curves of $f$, one needs only understand the finitely many critical level curves of $f$. In this paper, we show that in fact such a function $f$ is determined not just geometrically but conformally by the configuration of critical level curves. That is, if $f_1$ and $f_2$ have the same configuration of critical level curves, then there is a conformal map $ϕ$ such that $f_1\equiv f_2\circϕ$. We then show that every configuration of critical level curves which could come from an analytic function is instantiated by a polynomial. We also include a new proof of a theorem of Bôcher (which is an extension of the Gauss--Lucas theorem to rational functions) using level curves.

math.CV↗

arXiv2013

Notes on the Level Curves of a Meromorphic Function

The subject of this paper is the bounded level curves of a meromorphic function $f$ with domain $G$ such that each component of $\partial{G}$ consists of a level curve of $f$. (A primary example of such a function being a ratio of finite Blaschke products of different degrees, with domain $\mathbb{D}$.) We will first prove several facts about a single bounded level curve of a $f$ in isolation from the other level curves of $f$. We will then study how the level curves of $f$ lie with respect to each other. It is natural to expect that the sets $\{z:|f(z)|=ε\}$ vary continuously as $ε$ varies. We will make this notion explicit, and use this continuity to prove several results about the bounded level curves of $f$. It is well known that if $z_0$ is a zero or a pole of $f$, then $f$ is conformally equivalent to the function $z\mapsto{z^k}$ (for some $k\in\mathbb{Z}$) in a neighborhood of $z_0$. We generalize this fact by finding a natural decomposition of $G$ into finitely many sub-regions (also bounded by level curves of $f$), on each of which $f$ is conformally equivalent to $z\mapsto{z^k}$ (for some $k\in\mathbb{Z}$). Also included is a new proof, using level curves, of the Gauss--Lucas theorem that the critical points of a polynomial are contained in the convex hull of the polynomial's zeros.

math.CV↗