Related papers: Necessary Conditions for Single-Critical-Point Hig…

We give a direct algebraic proof of the necessity direction in the single-point higher-order Szeg\H{o} sum rules on the unit circle for $m=1,2,3$. More precisely, for $H_m(e^{i\theta})=(1-\cos\theta)^m$, we show that…

We prove the following higher-order Szego theorems: if a measure on the unit circle has absolutely continuous part $w(\theta)$ and Verblunsky coefficients $\alpha$ with square-summable variation, then for any positive integer $m$, $\int…

In this paper we prove Szeg\H{o}'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,…

This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $d\mu = w(\theta) \frac{d\theta}{2\pi} + d\mu_s$ with Verblunsky coefficients…

We consider probability measures, $d\mu=w(\theta) \f{d\theta}{2\pi} +d\mu_\s$, on the unit circle, $\partial\bbD$, with Verblunsky coefficients, $\{\alpha_j\}_{j=0}^\infty$. We prove for $\theta_1\neq\theta_2$ in $[0,2\pi)$ and…

Let $\fre\subset\bbR$ be a finite union of disjoint closed intervals. We study measures whose essential support is $\fre$ and whose discrete eigenvalues obey a 1/2-power condition. We show that a Szeg\H{o} condition is equivalent to \[…

The classical Szeg\H{o}-Verblunsky theorem relates integrability of the logarithm of the absolutely continuous part of a probability measure on the circle to square summability of the sequence of recurrence coefficients for the orthogonal…

A Jacobi matrix with $a_n\to 1$, $b_n\to 0$ and spectral measure $\nu'(x)dx + d\nu_{sing}(x)$ satisfies the Szeg\H o condition if $\int_{0}^\pi \ln \bigl[ \nu'(2\cos\theta) \bigr] d\theta$ is finite. We prove that if $a_n = 1 + \frac…

A new operator-level necessity result for the Chapman--Enskog expansion is established: in closed and unforced kinetic systems, the $O(\varepsilon)$ deviatoric stress arises if and only if the first Chapman--Enskog correction $f^{(1)}$ is…

We extend a higher-order sum rule proved by B. Simon to matrix valued measures on the unit circle and their matrix Verblunsky coefficients.

We establish versions of Szeg\H{o}'s distance formula and Widom's theorem on invertibility of (a family of) Toeplitz operators in a class of finite codimension subalgebras of uniform algebras, obtained by imposing a finite number of linear…

Models of self-organized criticality, which can be described as singular diffusions with or without (multiplicative) Wiener forcing term (as e.g. the Bak/Tang/Wiesenfeld- and Zhang-models), are analyzed. Existence and uniqueness of…

The critcal exponent $\omega$ is evaluated at $O(1/N)$ in $d$-dimensions in the Gross-Neveu model using the large $N$ critical point formalism. It is shown to be in agreement with the recently determined three loop $\beta$-functions of the…

The Szeg\H{o} limit theorem by Fedele and Gebert for matrices of the type identity minus Hankel matrix is proved for the special case $1-\frac{\beta}{\pi}H_{N,\alpha}$ where $H_{N,\alpha}$ is the $N\times N$-Hilbert matrix,…

The classical Szeg\"{o}--Kolmogorov Prediction Theorem gives necessary and sufficient condition on a weight $w$ on the unite cirlce $T$ so that the exponentials with positive integer frequences span the weighted space $L^2(T,w)$. We…

Given a measure $\mu$ on the unit sphere $\partial\mathbb{B}^d$ in $\mathbb{C}^d$ with Lebesgue decomposition ${\rm d} \mu = w \, {\rm d} \sigma + {\rm d} \mu_s$, with respect to the rotation-invariant Lebesgue measure $\sigma$ on $\partial…

Let $f$ be a function on the unit circle and $D_n(f)$ be the determinant of the $(n+1)\times (n+1)$ matrix with elements $\{c_{j-i}\}_{0\leq i,j\leq n}$ where $c_m =\hat f_m\equiv \int e^{-im\theta} f(\theta) \f{d\theta}{2\pi}$. The sharp…

Evaluation complexity for convexly constrained optimization is considered and it is shown first that the complexity bound of $O(\epsilon^{-3/2})$ proved by Cartis, Gould and Toint (IMAJNA 32(4) 2012, pp.1662-1695) for computing an…

In this paper we consider the higher order Lioville-type equation $(-\Delta)^{m} u=\rho^{2m} V(x) e^{u}$ in $\Omega\subseteq\mathbb{R}^{2m}$ with $V\neq0$ a given smooth potential, $\rho\in\mathbb{R}^{+}$ a small parameter which tends to…

Using recent developments on the theory of locally decodable codes, we prove that the critical size for Szemer\'edi's theorem with random differences is bounded from above by $N^{1-\frac{2}{k} + o(1)}$ for length-$k$ progressions. This…