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Meredith Sargent

Publications and source records attributed to Meredith Sargent.

6 recordsLinked to original sources
arXiv2022

An Optimal Approximation Problem For Free Polynomials

Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial $f$ in $d$ freely noncommuting arguments, find a free polynomial $p_n$, of degree at most $n$, to minimize $c_n := \|p_nf-1\|^2$. (Here the norm is the $\ell^2$ norm on coefficients.) We show that $c_n\to 0$ if and only if $f$ is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the $d$-shift.

math.FA↗
arXiv2021

Zero-free regions near a line

We analyze metrics for how close an entire function of genus one is to being real rooted. These metrics arise from truncated Hankel matrix positivity-type conditions built from power series coefficients at each real point. Specifically, if such a function satisfies our positivity conditions and has well-spaced zeros, we show that all of its zeros have to (in some explicitly quantified sense) be far away from the real axis. The obvious interesting example arises from the Riemann zeta function, where our positivity conditions yield a family of relaxations of the Riemann hypothesis. One might guess that as we tighten our relaxation, the zeros of the zeta function must be close to the critical line. We show that the opposite occurs: any potential complex zeros are forced to be farther and farther away from the critical line.

math.CV↗
arXiv2020

Optimal approximants and orthogonal polynomials in several variables

We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

math.CV↗
arXiv2018

Escaping nontangentiality: Towards a controlled tangential amortized Julia-Carathéodory theory

Let $f: D \rightarrow Ω$ be a complex analytic function. The Julia quotient is given by the ratio between the distance of $f(z)$ to the boundary of $Ω$ and the distance of $z$ to the boundary of $D.$ A classical Julia-Carathéodory type theorem states that if there is a sequence tending to $τ$ in the boundary of $D$ along which the Julia quotient is bounded, then the function $f$ can be extended to $τ$ such that $f$ is nontangentially continuous and differentiable at $τ$ and $f(τ)$ is in the boundary of $Ω.$ We develop an extended theory when $D$ and $Ω$ are taken to be the upper half plane which corresponds to amortized boundedness of the Julia quotient on sets of controlled tangential approach, so-called $λ$-Stolz regions, and higher order regularity, including but not limited to higher order differentiability, which we measure using $γ$-regularity. Applications are given, including perturbation theory and moment problems.

math.FA↗
arXiv2018

Carlson's Theorem for Different Measures

We use an observation of Bohr connecting Dirichlet series in the right half plane $\mathbb{C}_+$ to power series on the polydisk to interpret Carlson's theorem about integrals in the mean as a special case of the ergodic theorem by considering any vertical line in the half plane as an ergodic flow on the polytorus. Of particular interest is the imaginary axis because Carlson's theorem for Lebesgue measure does not hold there. In this note, we construct measures for which Carlson's theorem does hold on the imaginary axis for functions in the Dirichlet series analog of the disk algebra $\mathcal{A}(\mathbb{C}_+)$.

math.CV↗