arXiv subjects
Publications and source records attributed to Alex Bergman.
We use a lifting trick to show that the Beurling-Malliavin multiplier theorem extends to radial functions in several variables in a straightforward way. This simplifies an argument of Vasilyev and also answers a question of Vasilyev on the Cartwright version of the theorem.
A meromorphic inner function is a bounded holomorphic function in the upper half-plane which is unimodular on the real line and extends to a meromorphic function in the whole complex plane. The argument of a meromorphic inner function on the real line is a strictly increasing function. It turns out that it is important for many problems in function theory to approximate an arbitrary increasing function, $f$, by the argument of a meromorphic inner function. Depending on desired approximation this is a delicate problem. In this paper consider the case when $f$ satisfies a doubling condition. We give two applications of our main result. The first is a sufficient density condition for a set $\Lambda$ to be a zero set for a Toeplitz kernel with real analytic and unimodular symbol. Our second application is to describe a class of admissible Beurling-Malliavin majorants in model spaces. The generality considered here lets us treat most cases of model spaces generated by meromorphic one-component inner functions.
In this paper we provide a far-reaching generalization of the existent results about invariant subspaces of the differentiation operator $D=\frac{\partial}{\partial t}$ on $C^\infty(0,1)$ and the Volterra operator $Vf(t)=\int_0^tf(s)ds$, on $L^2(0,1)$. We use an abstract approach to study invariant subspaces of pairs $D,V$ with $DV=I$, where $V$ is compact and quasi-nilpotent and $D$ is unbounded densely defined and closed on the same Hilbert space. Our results cover many differential operators, like Schr\"odinger operators and a large class of other canonical systems, as well as the so-called compact self-adjoint operators with removable spectrum recently studied by Baranov and Yakubovich. Our methods are based on a model for such pairs which involves de Branges spaces of entire functions and plays a crucial role in the development. However, a number of difficulties arise from the fact that our abstract operators do not necessarily identify with the usual operators on such spaces, but with rank one perturbations of those, which, in terms of invariant subspaces creates a number of challenging problems.
Let $f$ be an entire function of finite exponential type less than or equal to $\sigma$ which is bounded by $1$ on the real axis and satisfies $f(0) = 1$. Under these assumptions H\"ormander showed that $f$ cannot decay faster than $\cos(\sigma x)$ on the interval $(-\pi/\sigma,\pi/\sigma)$. We extend this result to the setting of de Branges spaces with cosine replaced by the real part of the associated Hermite-Biehler function. We apply this result to study the point evaluation functional and associated extremal functions in de Branges spaces (equivalently in model spaces generated by meromorphic inner functions) generalizing some recent results of Brevig, Chirre, Ortega-Cerd\`a, and Seip.
Let $f$ be a holomorphic self-map of the unit disc. We show that if $\log (1-\lvert f(z) \rvert)$ is integrable on a sub-arc of the unit circle, $I$, then the set of points where the function f has finite Carath\'eodory angular derivative on I is a countable union of Beurling-Carleson sets of finite entropy. Conversely, given a countable union of Beurling-Carleson sets, $E$, we construct a holomorphic self-map of the unit disc, $f$, such that the set of points where the function has finite Carath\'eodory angular derivative is equal to $E$ and $\log(1-\lvert f(z) \rvert)$ is integrable on the unit circle. Our main technical tools are the Aleksandrov disintegration Theorem and a characterization of countable unions of Beurling-Carleson sets due to Makarov and Nikolski.
Estimating the overall user experience (UX) on a device is a common challenge faced by manufacturers. Today, device makers primarily rely on microbenchmark scores, such as Geekbench, that stress test specific hardware components, such as CPU or RAM, but do not satisfactorily capture consumer workloads. System designers often rely on domain-specific heuristics and extensive testing of prototypes to reach a desired UX goal, and yet there is often a mismatch between the manufacturers' performance claims and the consumers' experience. We present our initial results on predicting real-life experience on laptops from their hardware specifications. We target web applications that run on Chromebooks (ChromeOS laptops) for a simple and fair aggregation of experience across applications and workloads. On 54 laptops, we track 9 UX metrics on common end-user workloads: web browsing, video playback and audio/video calls. We focus on a subset of high-level metrics exposed by the Chrome browser, that are part of the Web Vitals initiative for judging the UX on web applications. With a dataset of 100K UX data points, we train gradient boosted regression trees that predict the metric values from device specifications. Across our 9 metrics, we note a mean $R^2$ score (goodness-of-fit on our dataset) of 97.8% and a mean MAAPE (percentage error in prediction on unseen data) of 10.1%.
We study the problem of characterizing the cyclic vectors in de Branges-Rovnyak spaces. Based on a description of the invariant subspaces we show that the difficulty lies entirely in understanding the subspace $(aH^{2})^{\perp}$ and give a complete function theoretic description of the cyclic vectors in the case $\dim (aH^{2})^{\perp} < \infty$. Incidentally, this implies analogous results for certain generalized Dirichlet spaces $\mathcal{D}(\mu)$. Most of our attention is directed to the infinite case where we relate the cyclicity problem to describing the exposed points of $H^{1}$ and provide several sufficient conditions. A necessary condition based on the Aleksandrov-Clark measures of $b$ is also presented.
We consider weighted Chebyshev polynomials on the unit circle corresponding to a weight of the form $(z-1)^s$ where $s>0$. For integer values of $s$ this corresponds to prescribing a zero of the polynomial on the boundary. As such, we extend findings of Lachance, Saff and Varga, to non-integer $s$. Using this generalisation, we are able to relate Chebyshev polynomials on lemniscates and other, more established, categories of Chebyshev polynomials. An essential part of our proof involves the broadening of the Erd\H{o}s--Lax inequality to encompass powers of polynomials. We believe that this particular result holds significance in its own right.