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Brian Simanek

Publications and source records attributed to Brian Simanek.

At least 37 records · Page 2Linked to original sources
arXiv2019

Spectra of Cayley graphs of the lamplighter group and random Schrodinger operators

We show that the lamplighter group L has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is the first example of a group with infinitely many gaps in the spectrum of its Cayley graph. The result is obtained by a careful study of spectral properties of a one-parametric family of convolution operators on L. Our results show that the spectrum is a pure point spectrum for each value of the parameter, the eigenvalues are solutions of algebraic equations involving Chebyshev polynomials of the second kind, and the topological structure of the spectrum makes a bifurcation when the parameter passes the points 1 and -1.

math.GT↗
arXiv2019

$H^\infty$-calculus for semigroup generators on BMO

We prove that the negative infinitesimal generator $L$ of a semigroup of positive contractions on $L^\infty$ has a bounded $H^\infty(S_η^0)$-calculus on the associated Poisson semigroup-BMO space for any angle $η>π/2$, provided the semigroup satisfies Bakry-Emry's $Γ_2 $ criterion. Our arguments only rely on the properties of the underlying semigroup and works well in the noncommutative setting. A key ingredient of our argument is a quasi monotone property for the subordinated semigroup $T_{t,α}=e^{-tL^α},0<α<1$, that is proved in the first half of the article.

math.FA↗
arXiv2018

A maximal energy pointset configuration problem

We consider the extremal pointset configuration problem of maximizing a kernel-based energy subject to the geometric constraints that the points are contained in a fixed set, the pairwise distances are bounded below, and that every closed ball of fixed radius contains at least one point. We also formulate an extremal density problem, whose solution provides an upper bound for the pointset configuration problem in the limit as the number of points tends to infinity. Existence of solutions to both problems is established and the relationship between the parameters in the two problems is studied. Several examples are studied in detail, including the density problem for the $d$-dimensional ball and sphere, where the solution can be computed exactly using rearrangement inequalities. We develop a computational method for the density problem that is very similar to the Merriman-Bence-Osher (MBO) diffusion-generated method. The method is proven to be increasing for all non-stationary iterations and is applied to study more examples.

math.OC↗
arXiv2018

Applications of a New Formula for OPUC with Periodic Verblunsky Coefficients

We find a new formula for the orthonormal polynomials corresponding to a measure mu on the unit circle whose Verblunsky coefficients are periodic. The formula is presented using the Chebyshev polynomials of the second kind and the discriminant of the periodic sequence. We present several applications including a resolution of a problem suggested by Simon in 2006 regarding the existence of singular points in the bands of the support of the measure and a universality result at all points of the essential support of mu.

math.CA↗
arXiv2017

Universality at an Endpoint for Orthogonal Polynomials with Geronimus-Type Weights

We provide a new closed form expression for the Geronimus polynomials on the unit circle and use it to obtain new results and formulas. Among our results is a universality result at an endpoint of an arc for polynomials orthogonal with respect to a Geronimus type weight on an arc of the unit circle. The key tool is a formula of McLaughlin for powers of a two-by-two matrix, which we use to derive convenient formulas for Geronimus polynomials.

math.CA↗
arXiv2017

Asymptotics for polynomials orthogonal in an indefinite metric

We continue studying polynomials generated by the Szegő recursion when a finite number of Verblunsky coefficients lie outside the closed unit disk. We prove some asymptotic results for the corresponding orthogonal polynomials and then translate them to the real line to obtain the Szegő asymptotics for the resulting polynomials. The latter polynomials give rise to a non-symmetric tridiagonal matrix but it is a finite-rank perturbation of a symmetric Jacobi matrix.

math.CA↗
arXiv2017

Torsional Rigidity and Bergman Analytic Content of Simply Connected Regions

We exploit the equality of Bergman analytic content and torsional rigidity of a simply connected domain to develop a new method for calculating these quantities. This method is particularly suitable for the case when the region in question is a polygon. A large number of examples are computed in explicit detail to demonstrate the utility of our ideas.

math.CA↗
arXiv2016

Two Universality Results for Polynomial Reproducing Kernels

We prove two new universality results for polynomial reproducing kernels of compactly supported measures. The first applies to measures on the unit circle with a jump and a singularity in the weight at $1$ and the second applies to area-type measures on a certain disconnected polynomial lemniscate. In both cases, we apply methods developed by Lubinsky to obtain our results.

math.CA↗
arXiv2016

On Szegő's theorem for a nonclassical case

In this paper we prove Szegő's Theorem for the case when a finite number of Verblunsky coefficients lie outside the closed unit disk. Although a form of this result was already proved by A.L. Sakhnovich, we use a very different method, which shows that the OPUC machinery can still be applied to deal with such nonclassical cases. The basic tool we use is Khrushchev's formula that in the classical case relates the absolutely continuous part of the measure and the $N$-th iterate of the Schur algorithm. It is noteworthy that Khrushchev's formula makes the proof short and extremely transparent. Also, we discuss Verblunsky's theorem for the case in question.

math.CA↗
arXiv2016

Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems

We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzsch-type theorems describing where they accumulate. As a consequence, we obtain detailed information regarding zeros of reproducing kernels in weighted spaces of analytic functions.

math.CA↗
arXiv2016

Relative Asymptotics for General Orthogonal Polynomials

We study right limits of the Bergman Shift matrix. Our results have applications to ratio asymptotics, weak asymptotic measures, relative asymptotics, and zero counting measures of the orthogonal and orthonormal polynomials.

math.CA↗
arXiv2016

An Electrostatic Interpretation of the Zeros of Paraorthogonal Polynomials on the Unit Circle

We show that if m is a probability measure with infinite support on the unit circle having no singular component and a differentiable weight, then the corresponding paraorthogonal polynomial P_n(z;B) solves an explicit second order linear differential equation. We also show that if T and B are distinct, then the pair {P_n(z;B),P_n(z;T)} solves an explicit first order linear system of differential equations. One can use these differential equations to deduce that the zeros of every paraorthogonal polynomial mark the locations of a set of particles that are in electrostatic equilibrium with respect to a particular external field.

math.CA↗
arXiv2015

Extremal Polarization Configurations for Integrable Kernels

Our main result shows that if a lower-semicontinuous kernel K satisfies some mild additional hypotheses, then asympotitically polarization optimal configurations are precisely those that are asymptotically distributed according to the equilibrium measure for the corresponding minimum energy problem.

math.CA↗
arXiv2014

Periodic Discrete Energy for Long-Range Potentials

We consider periodic energy problems in Euclidean space with a special emphasis on long-range potentials that cannot be defined through the usual infinite sum. One of our main results builds on more recent developments of Ewald summation to define the periodic energy corresponding to a large class of long-range potentials. Two particularly interesting examples are the logarithmic potential and the Riesz potential when the Riesz parameter is smaller than the dimension of the space. For these examples, we use analytic continuation methods to provide concise formulas for the periodic kernel in terms of the Epstein Hurwitz Zeta function. We apply our energy definition to deduce several properties of the minimal energy including the asymptotic order of growth and the distribution of points in energy minimizing configurations as the number of points becomes large. We conclude with some detailed calculations in the case of one dimension, which shows the utility of this approach.

math-ph↗
arXiv2014

The Bergman Shift Operator on Polynomial Lemniscates

We investigate the relationship between the Bergman shift operator and the support of the corresponding measure. We pay special attention to the situation when the measure of orthogonality is concentrated on a polynomial lemniscate. As an application of our new results, we obtain a ratio asymptotic result for a wide variety of measures supported on polynomial lemniscates.

math.CA↗
arXiv2013

Ratio Asymptotics, Hessenberg Matrices, and Weak Asymptotic Measures

We discuss the relationship between ratio asymptotics for general orthogonal polynomials and the asymptotics of the associated Bergman shift operator. More specifically, we consider the case in which a measure is supported on an infinite compact subset of the complex plane. We show that there is a straightforward connection between the corresponding orthonormal polynomials exhibiting ratio asymptotics and the corresponding Bergman shift operator being asymptotically Toeplitz. We also discuss a connection to the weak asymptotics of the measures derived from the orthonormal polynomials.

math.CA↗