On the factorization of matrices

Analysis Mathematica, 1977

International Mathematics Research Notices, 2018

We show that universality limits and other bounds imply pointwise asymptotics for orthonormal polynomials at the endpoints of the interval of orthonormality. As a consequence, we show that if $\mu $ is a regular measure supported on $\left [ -1,1\right ] $, and in a neighborhood of 1, $ \mu $ is absolutely continuous, while for some $\alpha>-1$, $\mu ^{\prime }\left ( t\right ) =h\left ( t\right ) \left ( 1-t\right )^{\alpha }$, where $ h\left ( t\right ) \rightarrow 1$ as $t\rightarrow 1-$, then the corresponding orthonormal polynomials $\left \{ p_{n}\right \} $ satisfy the asymptotic $$ \lim_{n\rightarrow \infty }\frac{p_{n}\left( 1-\frac{z^{2}}{2n^{2}}\right) }{ p_{n}\left( 1\right) }=\frac{J_{\alpha }^{\ast }\left( z\right) }{J_{\alpha }^{\ast }\left( 0\right) } $$uniformly in compact subsets of the plane. Here $J_{\alpha }^{\ast }\left ( z\right ) =J_{\alpha }\left ( z\right ) /z^{\alpha }$ is the normalized Bessel function of order $\alpha $. These are by far the most gene...

Advances in Mathematics, 2019

There has been considerable recent literature connecting Poncelet's theorem to ellipses, Blaschke products and numerical ranges, summarized, for example, in the recent book [16]. We show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and, in turn, can provide new insights to the theory of OPUC.

2004

Journal of Mathematical Analysis and Applications, 1995

Dedicated with respect and a ection to Professor M. A. Kaashoek on the occasion of his 60-th birthday We study connections between operator theoretic properties of Toeplitz operators acting on suitable Besikovitch spaces and factorizations of their symbols which are matrix valued almost periodic functions of several real variables. Among other things, we establish the existence of a twisted canonical factorization for locally sectorial symbols, and characterize one-sided invertibility of Toeplitz operators in terms of their symbols. In addition, we study stability of factorizations, and factorizations of hermitian valued almost periodic matrix functions of several variables.

Journal of Fourier Analysis and Applications, 2002

×ØÖ Øº This paper completes the proof of the L p -boundedness of bilinear operators associated to nonsmooth symbols or multipliers begun in Part I, our companion paper , by establishing the corresponding L p -boundedness of time-frequency paraproducts associated with tiles in phase plane. The affine invariant structure of such operators in conjunction with the geometric properties of the associated phase-plane decompositions allow Littlewood-Paley techniques to be applied locally, ie. on trees. Boundedness of the full time-frequency paraproduct then follows using 'almost orthogonality' type arguments relying on estimates for tree-counting functions together with decay estimates.

Complex Analysis and Operator Theory, 2008

In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space L 2 (R n ). Our approach applies more generally to families of n arbitrary commuting unitary operators in a complex Hilbert space H, or equivalent the spectral theory of a unitary representation U of the rank-n lattice Z n in R n . Starting with a non-zero vector ψ ∈ H, we look for relations among the vectors in the cyclic subspace in H generated by ψ. Since these vectors {U (k)ψ|k ∈ Z n } involve infinite "linear combinations," the problem arises of giving geometric characterizations of these non-trivial linear relations. A special case of the problem arose initially in work of Kolmogorov under the name L 2 -independence. This refers to infinite linear combinations of integral translates of a fixed function with l 2 -coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic integrals. . Primary 47B40, 47B06, 06D22, 62M15; Secondary 42C40, 62M20.

Journal d'Analyse Mathématique, 2003

Let E be a homogeneous compact set, for instance a Cantor set of positive length. Further let σ be a positive measure with supp(σ) = E. Under the condition that the absolutely continuous part of σ satisfies a Szegö-type condition we give an asymptotic representation, on and off the support, for the polynomials orthonormal with respect to σ. For the special case that E consists of a finite number of intervals and that σ has no singular component this is a nowaday well known result of Widom. If E = [a, b] it becomes a classical result due to Szegö and in case that there appears in addition a singular component, it is due to Kolmogorov-Krein. In fact the results are presented for the more general case that the orthogonality measure may have a denumerable set of mass-points outside of E which are supposed to accumulate on E only and to satisfy (together with the zeros of the associated Stieltjes function) the free-interpolation Carleson-type condition. Up to the case of a finite number of mass points this is even new for the single interval case. Furthermore, as a byproduct of our representations, we obtain that the recurrence coefficients of the orthonormal polynomials behave asymptotically almost periodic. Or in other words the Jacobi matrices associated with the above discussed orthonormal polynomials are compact perturbations of a one-sided restriction of almost periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory of Hardy spaces of character-automorphic functions and forms on Riemann surfaces of Widom type, we use also some ideas of scattering theory for one-dimensional Schrödinger equations.

Journal of Approximation Theory, 2008

We apply universality limits to asymptotics of spacing of zeros fx kn g of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result is lim n!1 x kn x k+1;n K n (x kn ; x kn) = 1 under minimal hypotheses on the weight, withKn denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the di¤erentiated kernels K

Journal of Computational and Applied Mathematics, 1999

Let f n g 1 n=1 be a sequence of points in the open unit disk in the complex plane and let B 0 = 1 and B n (z) = n Y k=0 k j k j k ? z 1 ? k z ; n = 1; 2; : : :;