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Thomas Ransford

Publications and source records attributed to Thomas Ransford.

At least 19 recordsLinked to original sources
arXiv2026

A functional inequality related to Domar's uniform boundedness theorem

We study the functional inequality \[ f(r+s)\le g(r)+\alpha f(s) \quad(r,s>0). \] Here $g:(0,\infty)\to[0,\infty)$ is a given decreasing function, $\alpha$ is a constant such that $0<\alpha<1$, and the problem is to determine whether the family of decreasing functions $f:(0,\infty)\to[0,\infty)$ that satisfy this inequality is bounded above by some finite function on $(0,\infty)$ and, if so, to find bounds for this function. We present a solution to this problem, and use it to give a new proof of a theorem of Domar on the uniform boundedness of certain families of subharmonic functions, in addition obtaining explicit bounds.

math.CA↗
arXiv2026

A pathological de Branges-Rovnyak space

We construct a de Branges-Rovnyak space containing the polynomials as a dense subspace, together with a function in the space that lies outside the closed linear span of its Taylor partial sums. The space has the further property that the monomials are bounded in norm. The construction is carried out by reducing it to an elementary problem about series, which we then solve.

math.CV↗
arXiv2026

On the coefficient formula for de Branges-Rovnyak norms

Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\infty$, and let $\phi=b/a$, where $a$ is the Pythagorean mate of $b$. It is known that, if $f$ is a function holomorphic on a neighbourhood of the closed unit disk, then it belongs to $\mathcal{H}(b)$, and its norm in $\mathcal{H}(b)$ can be expressed in terms of the Taylor coefficients of $f$ and $\phi$ via the formula \[ \|f\|_{\mathcal{H}(b)}^2=\sum_{m\ge0}|\hat{f}(m)|^2 +\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2. \] However, the formula can break down for some other $f\in\mathcal{H}(b)$. In this article we extend the scope of the formula to all $f\in H^2$ for which the right-hand side is finite, provided that either $\phi\in H^2$ or $\phi$ is rational. If merely $\phi\in H^p$ for some $p\in(0,2]$, then the formula still holds provided that, in addition, $\sum_{m\ge0}m^{2/p-1}|\hat{f}(m)|^2<\infty$. We also establish a limit-form of the formula that is valid for all non-extreme $b$ and all $f\in\mathcal{H}(b)$.

math.CV↗
arXiv2026

A proof of Esterle's conjecture on negative powers of Hilbert-space contractions

We establish the following result, confirming a conjecture of Jean Esterle. For each closed subset $E$ of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\to\infty$ with the following property: if $T$ is a contraction on a Hilbert space such that $\sigma(T)\subset E$ and $\|T^{-n}\|=O(u_n)$ as $n\to\infty$, then $T$ is a unitary operator. A key tool used in the proof is a result generalizing the well-known fact that closed subsets $E$ of the real axis of Lebesgue measure zero are removable for bounded holomorphic functions. We show that such sets remain removable even for certain unbounded holomorphic functions of moderate growth near $E$, where the notion of `moderate' depends on $E$.

math.FA↗
arXiv2026

A Schwarz-Jack lemma, circularly symmetric domains and numerical ranges

We prove a Schwarz-Jack lemma for holomorphic functions on the unit disk with the property that their maximum modulus on each circle about the origin is attained at a point on the positive real axis. With the help of this result, we establish monotonicity and convexity properties of conformal maps of circularly symmetric and bi-circularly symmetric domains. As an application, we give a new proof of Crouzeix's theorem that the numerical range of any $2\times 2$ matrix is a $2$-spectral set for the matrix. Unlike other proofs, our approach does not depend on the explicit formula for the conformal mapping of an ellipse onto the unit disk.

math.CV↗
arXiv2025

A Cramér-Wold theorem for mixtures

We show how a Cramér-Wold theorem for a family of multivariate probability distributions can be used to generate a similar theorem for mixtures (convex combinations) of distributions drawn from the same family. Using this abstract result, we establish a Cramér-Wold theorem for mixtures of multivariate Gaussian distributions. According to this theorem, two such mixtures can be distinguished by projecting them onto a certain predetermined finite set of lines, the number of lines depending only on the total number Gaussian distributions involved and on the ambient dimension. A similar result is also obtained for mixtures of multivariate $t$-distributions.

math.PR↗
arXiv2025

Double-layer potentials, configuration constants and applications to numerical ranges

Given a compact convex planar domain $Ω$ with non-empty interior, the classical Neumann's configuration constant $c_{\mathbb{R}}(Ω)$ is the norm of the Neumann-Poincaré operator $K_Ω$ acting on the space of continuous real-valued functions on the boundary $\partial Ω$, modulo constants. We investigate the related operator norm $c_{\mathbb{C}}(Ω)$ of $K_Ω$ on the corresponding space of complex-valued functions, and the norm $a(Ω)$ on the subspace of analytic functions. This change requires introduction of techniques much different from the ones used in the classical setting. We prove the equality $c_{\mathbb{R}}(Ω) = c_{\mathbb{C}}(Ω)$, the analytic Neumann-type inequality $a(Ω) < 1$, and provide various estimates for these quantities expressed in terms of the geometry of $Ω$. We apply our results to estimates for the holomorphic functional calculus of operators on Hilbert space of the type $\|p(T)\| \leq K \sup_{z \in Ω} |p(z)|$, where $p$ is a polynomial and $Ω$ is a domain containing the numerical range of the operator $T$. Among other results, we show that the well-known Crouzeix-Palencia bound $K \leq 1 + \sqrt{2}$ can be improved to $K \leq 1 + \sqrt{1 + a(Ω)}$. In the case that $Ω$ is an ellipse, this leads to an estimate of $K$ in terms of the eccentricity of the ellipse.

math.FA↗
arXiv2025

Two statistical problems for multivariate mixture distributions

We address two important statistical problems: that of estimating mixtures of multivariate normal distributions and mixtures of $t$-distributions based on univariate projections, and that of quantifying a discrepancy between mixture distributions induced by two model-based clusterings. In the second problem, rather than introducing a direct metric on partitions, we propose a model-based distributional discrepancy between the fitted mixture distributions associated with two clusterings. The results are based on an earlier work of the authors, where it was shown that mixtures of multivariate Gaussian or $t$-distributions can be distinguished by projecting them onto a certain predetermined finite set of lines, the number of lines depending only on the total number of distributions involved and on the ambient dimension. We also compare our proposal with robust versions of the expectation-maximization method EM. In each case, we present algorithms for effecting the task, and compare them with existing methods by carrying out some simulations.

math.ST↗
arXiv2024

Contractivity of Möbius functions of operators

Let $T$ be a injective bounded linear operator on a complex Hilbert space. We characterize the complex numbers $λ,μ$ for which $(I+λT)(I+μT)^{-1}$ is a contraction, the characterization being expressed in terms of the numerical range of the possibly unbounded operator $T^{-1}$. When $T=V$, the Volterra operator on $L^2[0,1]$, this leads to a result of Khadkhuu, Zemánek and the second author, characterizing those $λ,μ$ for which $(I+λV)(I+μV)^{-1}$ is a contraction. Taking $T=V^n$, we further deduce that $(I+λV^n)(I+μV^n)^{-1}$ is never a contraction if $n\ge2$ and $λ\neμ$.

math.FA↗
arXiv2024

On the Crouzeix ratio for $N\times N$ matrices

The Crouzeix ratio $ψ(A)$ of an $N\times N$ complex matrix $A$ is the supremum of $\|p(A)\|$ taken over all polynomials $p$ such that $|p|\le 1$ on the numerical range of $A$. It is known that $ψ(A)\le 1+\sqrt{2}$, and it is conjectured that $ψ(A)\le 2$. In this note, we show that $ψ(A)\le C_N$, where $C_N$ is a constant depending only on $N$ and satisfying $C_N<1+\sqrt{2}$. The proof is based on a study of the continuity properties of the map $A\mapsto ψ(A)$.

math.FA↗
arXiv2024

Negative powers of Hilbert-space contractions

We show that, given a closed subset $E$ of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\to\infty$ with the following property: if $T$ is a Hilbert-space contraction such that $σ(T)\subset E$ and $\|T^{-n}\|=O(u_n)$ and rank$(I-T^*T)<\infty$, then $T$ is a unitary operator. We further show that the condition of measure zero is sharp.

math.FA↗
arXiv2023

On the Divergence of Taylor Series in de Branges-Rovnyak Spaces

It is known that there exist functions in certain de Branges--Rovnyak spaces whose Taylor series diverge in norm, even though polynomials are dense in the space. This is often proved by showing that the sequence of Taylor partial sums is unbounded in norm. In this note we show that it can even happen that the Taylor partial sums tend to infinity in norm. We also establish similar results for lower-triangular summability methods such as the Cesàro means.

math.CV↗
arXiv2023

Holomorphic motions, dimension, area and quasiconformal mappings

We describe the variation of the Minkowski, packing and Hausdorff dimensions of a set moving under a holomorphic motion, as well as the variation of its area. Our method provides a new, unified approach to various celebrated theorems about quasiconformal mappings, including the work of Astala on the distortion of area and dimension under quasiconformal mappings and the work of Smirnov on the dimension of quasicircles.

math.CV↗
arXiv2023

A quantitative Heppes Theorem and multivariate Bernoulli distributions

Using some extensions of a theorem of Heppes on finitely supported discrete probability measures, we address the problems of classification and testing based on projections. In particular, when the support of the distributions is known in advance (as for instance for multivariate Bernoulli distributions), a single suitably chosen projection determines the distribution. Several applications of these results are considered.

math.PR↗
arXiv2023

A Cramér-Wold theorem for elliptical distributions

According to a well-known theorem of Cramér and Wold, if $P$ and $Q$ are two Borel probability measures on $\mathbb{R}^d$ whose projections $P_L,Q_L$ onto each line $L$ in $\mathbb{R}^d$ satisfy $P_L=Q_L$, then $P=Q$. Our main result is that, if $P$ and $Q$ are both elliptical distributions, then, to show that $P=Q$, it suffices merely to check that $P_L=Q_L$ for a certain set of $(d^2+d)/2$ lines $L$. Moreover $(d^2+d)/2$ is optimal. The class of elliptical distributions contains the Gaussian distributions as well as many other multivariate distributions of interest. Our theorem contrasts with other variants of the Cramér-Wold theorem, in that no assumption is made about the finiteness of moments of $P$ and $Q$. We use our results to derive a statistical test for equality of elliptical distributions, and carry out a small simulation study of the test, comparing it with other tests from the literature. We also give an application to learning (binary classification), again illustrated with a small simulation

math.PR↗
arXiv2023

Summability and duality

We formalize the observation that the same summability methods converge in a Banach space $X$ and its dual $X^*$. At the same time we determine conditions under which these methods converge in the weak and weak*-topologies on $X$ and $X^*$ respectively. We also derive a general limitation theorem, which yields a necessary condition for the convergence of a summability method in $X$. These results are then illustrated by applications to a wide variety of function spaces, including spaces of continuous functions, Lebesgue spaces, the disk algebra, Hardy and Bergman spaces, the BMOA space, the Bloch space, and de Branges-Rovnyak spaces. Our approach shows that all these applications flow from just two abstract theorems.

math.FA↗
arXiv2023

Asymptotically equicontinuous sequences of operators and a Banach-Steinhaus type theorem

We introduce the notion of an asymptotically equicontinuous sequence of linear operators, and use it to prove the following result. If $X,Y$ are topological vector spaces, if $T_n,T:X\to Y$ are continuous linear maps, and if $D$ is a dense subset of $X$, then the following statements are equivalent: (i) $T_nx\to Tx$ for all $x\in X$, and (ii) $T_n x\to Tx$ for all $x\in D$ and the sequence $(T_n)$ is asymptotically equicontinuous.

math.FA↗