arXiv subjects

Wayne Lawton

Publications and source records attributed to Wayne Lawton.

13 recordsLinked to original sources

arXiv2021

An Explanation of Mellin's 1921 Paper

In 1921 Mellin published a Comptes Rendu paper computing the principal root of the polynomial $Z^n + x_1Z^{n_1} + \cdots + x_pZ^{n_p} - 1$ using hypergeometric functions of its coefficients $x_1,...,x_p.$ He used an integral transform nowadays bearing his name. Slightly over three pages, the paper is written in French in a terse style befitting the language. Unable to find an elementary explanation on the web or in a texbook, we wrote this expository article to make Mellin's landmark result accessible to interested people who are not experts in hypergeometric functions and complex analysis.

math.AG↗

arXiv2019

Spectral Factorization and Entire Functions

The Fejér-Riesz spectral factorization lemma, which represents a nonnegative trigonometric polynomial as the squared modulus of a trigonometric polynomial, was extended by Ahiezer to factorize certain entire functions and by Helson and Lowdenslager to factorize certain functions on compact connected abelian groups whose Pontryagin duals are equipped with a linear order. This paper relates these factorizations for archimedian orders using the theory of almost periodic functions.

math.FA↗

arXiv2019

Distribution of Small Values of Bohr Almost Periodic Functions with Bounded Spectrum

If $f$ is a nonzero Bohr almost periodic function on $\mathbb R$ with a bounded spectrum we prove there exist $C_f > 0$ and integer $n > 0$ such that for every $u > 0$ the mean measure of the set $\{\, x \, : \, |f(x)| < u \, \}$ is less than $C_f\, u^{1/n}.$ For trigonometric polynomials with $\leq n + 1$ frequencies we show that $C_f$ can be chosen to depend only on $n$ and the modulus of the largest coefficient of $f.$ We show this bound implies that the Mahler measure $M(h),$ of the lift $h$ of $f$ to a compactification $G$ of $\mathbb R,$ is positive and discuss the relationship of Mahler measure to the Riemann Hypothesis.

math.FA↗

arXiv2016

Refinable functions with PV dilations

A PV number is an algebraic integer $α$ of degree $d \geq 2$ all of whose Galois conjugates other than itself have modulus less than $1$. Erdös \cite{erdos} proved that the Fourier transform $\widehat φ,$ of a nonzero compactly supported scalar valued function satisfying the refinement equation $φ(x) = \frac{|α|}{2}φ(αx) + \frac{|α|}{2}φ(αx-1)$ with $PV$ dilation $α,$ does not vanish at infinity so by the Riemann-Lebesgue lemma $φ$ is not integrable. Dai, Feng and Wang \cite{daifengwang} extended his result to scalar valued solutions of $φ(x) = \sum_k a(k) φ(αx - τ(k))$ where $τ(k)$ are integers and $a$ has finite support and sums to $|α|$. In (\cite{lawton3}, Conjecture 4.2) we conjectured that their result holds under the weaker assumption that $τ$ has values in the ring of polynomials in $α$ with integer coefficients. This paper formulates a stronger conjecture and provides support for it based on a solenoidal representation of $\widehat φ,$ and deep results of Erdös and Mahler \cite{erdosmahler};Odoni \cite{odoni} that give lower bounds for the asymptotic density of integers represented by integral binary forms of degree $> 2;$degree $ = 2,$ respectively. We also construct an integrable vector valued refinable function with PV dilation.

math.GN↗

arXiv2015

Multiresolution Analyses on Quasilattices

We derive relations between geometric means of the Fourier moduli of a refinable distribution and of a related polynomial. We use Pisot-Vijayaraghavan numbers to construct families of one dimension quasilattices and multiresolution analyses spanned by distributions that are refinable with respect to dilation by the PV numbers and translation by quasilattice points. We conjecture that scalar valued refinable distributions are never integrable, construct piecewise constant vector valued refinable functions, and discuss multidimensional extensions.

math.CA↗

arXiv2013

Kicked-Harper model vs On-Resonance Double Kicked Rotor Model: From Spectral Difference to Topological Equivalence

Recent studies have established that, in addition to the well-known kicked Harper model (KHM), an on-resonance double kicked rotor model (ORDKR) also has Hofstadter's butterfly Floquet spectrum, with strong resemblance to the standard Hofstadter's spectrum that is a paradigm in studies of the integer quantum Hall effect. Earlier it was shown that the quasi-energy spectra of these two dynamical models (i) can exactly overlap with each other if an effective Planck constant takes irrational multiples of 2*pi and (ii) will be different if the same parameter takes rational multiples of 2*pi. This work makes some detailed comparisons between these two models, with an effective Planck constant given by 2*pi M/N, where M and N are coprime and odd integers. It is found that the ORDKR spectrum (with two periodic kicking sequences having the same kick strength) has one flat band and $N-1$ non-flat bands whose largest width decays in power law as K to the power of N+2, where K is a kicking strength parameter. The existence of a flat band is strictly proven and the power law scaling, numerically checked for a number of cases, is also analytically proven for a three-band case. By contrast, the KHM does not have any flat band and its band width scales linearly with K. This is shown to result in dramatic differences in dynamical behavior, such as transient (but extremely long) dynamical localization in ORDKR, which is absent in KHM. Finally, we show that despite these differences, there exist simple extensions of KHM and ORDKR (upon introducing an additional periodic phase parameter) such that the resulting extended KHM and ORDKR are actually topologically equivalent, i.e., they yield exactly the same Floquet-band Chern numbers and display topological phase transitions at the same kick strengths. A theoretical derivation of this topological equivalence is provided.

nlin.CD↗

arXiv2012

Spectral Factorization of Trigonometric Polynomials and Lattice Geometry

We formulate a conjecture concerning spectral factorization of a class of trigonometric polynomials of two variables and prove it for special cases. Our method uses relations between the distribution of values of a polynomial of two variables and the distributions of values of an associated family of polynomials of one variable. We suggest an approach to prove the full conjecture using relations between the distribution of values and the distribution of roots of polynomials.

math.NT↗

arXiv2012

Spectral Envelopes - A Preliminary Report

The spectral envelope S(F) of a subset of integers is the set of probability measures on the circle group that are weak star limits of squared moduli of trigonometric polynomials with frequencies in F. Fourier transforms of these measures are positive and supported in F - F but the converse generally fails. The characteristic function chiF of F is a binary sequence whose orbit closure gives a symbolic dynamical system O(F). Analytic properties of S(F) are related to dynamical properties of chiF. The Riemann-Lebesque lemma implies that if chiF is minimal, then S(F) is convex and hence S(F) is the closure of the convex hull of its extreme points Se(F). In this paper we (i) review the relationship between these concepts and the special case of the still open 1959 Kadison-Singer problem called Feichtinger's conjecture for exponential functions, (ii) partially characterize of elements in Se(F), for minimal chiF, in terms of ergodic properties of (O(F),lambda) where lambda is a shift invariant probability measure whose existence in ensured by the 1937 Krylov-Bogoyubov theorem, (iii) refine previous numerical studies of the Morse-Thue minimal binary sequence by exploiting a new MATLAB algorithm for computing smallest eigenvalues of 4,000,000 x 4,000,000 matrices, (iv) describe recent results characterizing S(F) for certain Bohr sets F related to quasicrystals, (v) extend these concepts to general discrete groups including those with Kazhdan's T-property, such as SL(n,Z), n > 2, which can be characterized by several equivalent properties such as: any sequence of positive definite functions converging to 1 uniformly on compact subsets converges uniformly. This exotic property may be useful to construct a counterexample to the generalization of Feichtinger's conjecture and hence to provide a no answer to the question of Kadison and Singer whcih they themselves tended to suspect.

math.FA↗

arXiv2012

Factorization of polynomials with analytic coefficients

We study monic univariate polynomials whose coefficients are analytic functions of a real variable and whose roots lie in a specified analytic curve. These include characteristic polynomials of unitary and hermitian matrices whose entries are analytic functions. We use a result of Newton to prove that every polynomial in such a class is a product of degree one polynomials in the class.

math.AG↗

arXiv2011

Spectral Factorization and Lattice Geometry

We obtain conditions for a trigonometric polynomial t of one variable to equal or be approximated by |p|^2 where p has frequencies in a Bohr set of integers obtained by projecting lattice points in the open planar region bounded by the lines y = alpha*x +- beta where |beta| leq 1/4 and alpha is either rational or irrational with Liouville-Roth constant larger than 2. We derive and use a generalization of the Fejer-Riesz spectral factorization lemma in one dimension, an approximate spectral factorization in two dimensions, the modular group action on the integer lattice, and Diophantine approximation.

math.NT↗

arXiv2010

The Feichtinger Conjecture for Exponentials

The Feichtinger conjecture for exponentials asserts that the following property holds for every fat Cantor subset B of the circle group: the set of restrictions to B of exponential functions can be covered by Riesz sets. In their seminal paper on the Kadison-Singer problem, Bourgain and Tzafriri proved that this property holds if the characteristic function of B has Sobolev regularity. Their probability based proof does not explicitly construct a Riesz cover. They also showed how to construct fat Cantor sets whose characteristic functions have Sobolev regularity. However, these fat Cantor sets are not convenient for numerical calculations. This paper addresses these concerns. It constructs a family of fat Cantor sets, parameterized by their Haar measure, whose characteristic functions have Sobolev regularity and their Fourier transforms are Riesz products. It uses these products to perform computational experiments that suggest that if the measure of one of these fat Cantor sets B is sufficiently close to one, then it may be possible to explicitly construct a Riesz cover for B using the Thue-Morse minimal sequence that arises in symbolic topological dynamics.

math.FA↗

arXiv2008

Spectral Relationships Between Kicked Harper and On-Resonance Double Kicked Rotor Operators

Kicked Harper operators and on-resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectrums of these operators. In this paper we apply C*-algebra methods to explain this resemblance. We show that each pair of corresponding operators belong to a common rotation C*-algebra B_α, prove that their spectrums are equal if αis irrational, and prove that the Hausdorff distance between their spectrums converges to zero as q increases if α= p/q with p and q coprime integers. Moreover, we show that corresponding operators in B_αare homomorphic images of mother operators in the universal rotation C*-algebra A_αthat are unitarily equivalent and hence have identical spectrums. These results extend analogous results for almost Mathieu operators. We also utilize the C*-algebraic framework to develop efficient algorithms to compute the spectrums of these mother operators for rational αand present preliminary numerical results that support the conjecture that their spectrums are Cantor sets if αis irrational. This conjecture for almost Mathieu operators, called the Ten Martini Problem, was recently proved after intensive efforts over several decades. This proof for the almost Mathieu operators utilized transfer matrix methods, which do not exist for the kicked operators. We outline a strategy, based on a special property of loop groups of semisimple Lie groups, to prove this conjecture for the kicked operators.

math-ph↗

arXiv2007

Proof of the Hyperplane Zeros Conjecture of Lagarias and Wang

We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics and Fourier analysis.

math.AG↗