Covering numbers and schlicht functions
We determine upper and lower bounds for the minimal number of balls of a given radius needed to cover the space of schlicht functions.
math.CV↗
arXiv subjects
Thomas Ransford
Publications and source records attributed to Thomas Ransford.
Approximation by random complex polynomials and random rational functions
We investigate random compact sets with random functions defined thereon, such as polynomials, rational functions, the pluricomplex Green function and the Siciak extremal function. One surprising consequence of our study is that randomness can be used to `improve' convergence for sequences of functions.
math.CV↗
Outer functions and uniform integrability
We show that, if $f$ is an outer function and $a\in[0,1)$, then the set of functions $\{\log |(f\circψ)^*|: ψ:\mathcal{D}\to\mathcal{D} \text{ holomorphic}, |ψ(0)|\le a\}$ is uniformly integrable on the unit circle. As an application, we derive a simple proof of the fact that, if $f$ is outer and $ϕ:\mathcal{D}\to\mathcal{D}$ is holomorphic, then $f\circϕ$ is outer.
math.CV↗
Cesàro summability of Taylor series in weighted Dirichlet spaces
We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,α)$-summable to the function in the norm of the space, provided that $α>1/2$. We further show that the constant $1/2$ is sharp, in marked contrast with the classical case of the disk algebra.
math.CV↗
Failure of Approximation of Odd Functions by Odd Polynomials
We construct a Hilbert holomorphic function space $H$ on the unit disk such that the polynomials are dense in $H$, but the odd polynomials are not dense in the odd functions in $H$. As a consequence, there exists a function $f$ in $H$ that lies outside the closed linear span of its Taylor partial sums $s_n(f)$, so it cannot be approximated by any triangular summability method applied to the $s_n(f)$. We also show that there exists a function $f$ in $H$ that lies outside the closed linear span of its radial dilates $f_r, ~r<1$.
math.FA↗
Crouzeix's Conjecture and related problems
In this paper, we establish several results related to Crouzeix's conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separated. This separation is measured by the so-called separation constant, which is defined in terms of the pseudohyperbolic metric. Moreover, we study general properties of related extremal functions and associated vectors. Throughout, compressions of the shift serve as illustrating examples which also allow for refined results.
math.FA↗
Continuity of capacity of a holomorphic motion
We study the behavior of various set-functions under holomorphic motions. We show that, under such deformations, logarithmic capacity varies continuously, while analytic capacity may not.
math.CV↗
Linear maps preserving inner functions
We show that, for many holomorphic function spaces on the unit disk, a continuous endomorphism that sends inner functions to inner functions is necessarily a weighted composition operator.
math.CV↗
One-box conditions for Carleson measures for the Dirichlet space
We give a simple proof of the fact that a finite measure $μ$ on the unit disk is a Carleson measure for the Dirichlet space if it satisfies the Carleson one-box condition $μ(S(I))=O(ϕ(|I|))$, where $ϕ:(0,2π]\to(0,\infty)$ is an increasing function such that $\int_0^{2π}(ϕ(x)/x)\,dx<\infty$. We further show that the integral condition on $ϕ$ is sharp.
math.CV↗
Outer functions and divergence in de Branges-Rovnyak spaces
In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function $f$ can be approximated in norm by its dilates $f_r(z):=f(rz)~(r<1)$, in other words, $\lim_{r\to1^-}\|f_r-f\|=0$. We construct a de Branges-Rovnyak space ${\mathcal H}(b)$ in which the polynomials are dense, and a function $f\in{\mathcal H}(b)$ such that $\lim_{r\to1^-}\|f_r\|_{{\mathcal H}(b)}=\infty$. The essential feature of our construction lies in the fact that $b$ is an outer function.
math.CV↗
A Gleason-Kahane-Żelazko theorem for the Dirichlet space
We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of weighted composition operators on the Dirichlet space as being exactly those linear maps that send nowhere-vanishing functions to nowhere-vanishing functions. We also investigate possible extensions to weighted Dirichlet spaces with superharmonic weights. As part of our investigation, we are led to determine which of these spaces contain functions that map the unit disk onto the whole complex plane.
math.FA↗
Remarks on the Crouzeix-Palencia proof that the numerical range is a $(1+\sqrt2)$-spectral set
Crouzeix and Palencia recently showed that the numerical range of a Hilbert-space operator is a $(1+\sqrt2)$-spectral set for the operator. One of the principal ingredients of their proof can be formulated as an abstract functional-analysis lemma. We give a new short proof of the lemma and show that, in the context of this lemma, the constant $(1+\sqrt2)$ is sharp.
math.FA↗
A uniform boundedness principle in pluripotential theory
For families of continuous plurisubharmonic functions we show that, in a local sense, separately bounded above implies bounded above.
math.CV↗
Tricomplex dynamical systems generated by polynomials of even degree
In this article, we give the exact interval of the cross section of the Multibrot sets generated by the polynomial $z^p+c$ where $z$ and $c$ are complex numbers and $p \geq 2$ is an even integer. Furthermore, we show that the same Multibrots defined on the hyperbolic numbers are always squares. Moreover, we give a generalized 3D version of the hyperbolic Multibrot set and prove that our generalization is an octahedron for a specific 3D slice of the tricomplex polynomial $η^p+c$ where $p \geq 2$ is an even integer.
math.GM↗
Cross-sections of multibrot sets
We identify the intersection of the multibrot set of $z^d+c$ with the rays ${\mathbb R}^+ω$, where $ω^{d-1}=\pm1$.
math.DS↗
Approximation in the closed unit ball
In this expository article, we present a number of classic theorems that serve to identify the closure in the sup-norm of various sets of Blaschke products, inner functions and their quotients, as well as the closure of the convex hulls of these sets. The results presented include theorems of Carathéodory, Fisher, Helson-Sarason, Frostman, Adamjan-Arov-Krein, Douglas-Rudin and Marshall. As an application of some of these ideas, we obtain a simple proof of the Berger-Stampfli spectral mapping theorem for the numerical range of an operator.
math.CV↗
Spectral sets for numerical range
We define and study a numerical-range analogue of the notion of spectral set. Among the results obtained are a positivity criterion and a dilation theorem, analogous to those already known for spectral sets. An important difference from the classical definition is the role played in the new definition by the base point. We present some examples to illustrate this aspect.
math.FA↗
Cyclicity in the harmonic Dirichlet space
The harmonic Dirichlet space $\cal{D} (\mathbb{T})$ is the Hilbert space of functions $f \in L^2(\mathbb{T})$ such that $$\|f\|_{\cal{D} (\mathbb{T})}^2 := \sum_{n\in\mathbb{Z}} (1+|n|)|\hat{f}(n)|^2 < \infty.$$ We give sufficient conditions for $f$ to be cyclic in $\cal{D} (\mathbb{T})$, in other words, for $\{ζ^nf(ζ):\ n\geq 0\}$ to span a dense subspace of $\cal{D} (\mathbb{T})$.
math.CV↗