Related papers: A matrix version of a higher-order Szeg{\"o} theor…
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…
Following the recent survey by the same author of Szeg\"o's theorem and orthogonal polynomials on the unit circle (OPUC) in the scalar case, we survey the corresponding multivariate prediction theory and matrix OPUC (MOPUC).
We introduce a theory of orthogonal polynomials on the unit sphere of the quaternions based on the notion of a $q$-positive measure (which originated in a work of Alpay, Colombo, the second author and Sabadini). The results we extend to…
We consider canonical systems and investigate the Szeg\H{o} class, which is defined via the finiteness of the associated entropy functional. Noting that the canonical system may be studied in a variety of gauges, we choose to work in the…
A sum rule is an identity connecting the entropy of a measure with coefficients involved in the construction of its orthogonal polynomials (Jacobi coefficients). Our paper is an extension of Gamboa, Nagel and Rouault (2016), where we have…
We use the large deviation approach to sum rules pioneered by Gamboa, Nagel and Rouault to prove higher order sum rules for orthogonal polynomials on the unit circle. In particular, we prove one half of a conjectured sum rule of Lukic in…
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…
We study the Case sum rules, especially $C_0$, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat's theorem to cases with an infinite point spectrum and a proof that…
We study the moment space corresponding to matrix measures on the unit circle. Moment points are characterized by non-negative definiteness of block Toeplitz matrices. This characterization is used to derive an explicit representation of…
By using the Szeg\H{o}'s transformation we deduce new relations between the recurrence coefficients for orthogonal polynomials on the real line and the Verblunsky parameters of orthogonal polynomials on the unit circle. Moreover, we study…
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 take the trace of Von-Neumann's ergodic theorem and get a trace formula of a unitary matrix family. It is an extension of Poisson summation formula in higher dimension. We also construct a family of crystalline measure with complex…
We apply a theorem of Geronimus to derive some new formulas connecting Schur functions with orthogonal polynomials on the unit circle. The applications include the description of the associated measures and a short proof of Boyd's result…
We put forward a proof of Solomon's rule, in terms of matrices, for multiplication in the descent algebra of the symmetric group. Our proof exploits the graphs that we can obtain from all the subsets of the set of transpositions,…
The theory of orthogonal polynomials on the unit circle (OPUC) dates back to Szeg\"o's work of 1915-21, and has been given a great impetus by the recent work of Simon, in particular his two-volume book [Si4], [Si5], the survey paper (or…
We prove that the Szeg\H{o} function, $D(z)$, of a measure on the unit circle is entire meromorphic if and only if the Verblunsky coefficients have an asymptotic expansion in exponentials. We relate the positions of the poles of $D(z)^{-1}$…
Extending earlier work of Killip-Simon and Simon-Zlatos, we obtain sum rules for Jacobi matrices in which the a.c. part of the spectral measure and the eigenvalues of the matrix appear on opposite sides of the equation. We use these to…
We generalize Bourgain's discretized sum-product theorem to matrix algebras.
The aim of the present work is to show that the results obtained earlier on the approximation of distributions of sums of independent summands by infinitely divisible laws may be transferred to the estimation of the closeness of…
Baiocchi et al. generalized a few years ago a classical theorem of Ingham and Beurling by means of divided differences. The optimality of their assumption has been proven by the third author of this note. The purpose of this note to extend…