An expansion for polynomials orthogonal over an analytic Jordan curve

We consider polynomials that are orthogonal over an analytic Jordan curve L with respect to a positive analytic weight, and show that each such polynomial of sufficiently large degree can be expanded in a series of certain integral transforms that converges uniformly in the whole complex plane. This expansion yields, in particular and simultaneously, Szego's classical strong asymptotic formula and a new integral representation for the polynomials inside L. We further exploit such a representation to derive finer asymptotic results for weights having finitely many singularities (all of algebraic type) on a thin neighborhood of the orthogonality curve. Our results are a generalization of those previously obtained in [7] for the case of L being the unit circle.Comment: 15 pages, 1 figur

Eigenvalue Bounds for Perturbations of Schrodinger Operators and Jacobi Matrices With Regular Ground States

We prove general comparison theorems for eigenvalues of perturbed Schrodinger operators that allow proof of Lieb--Thirring bounds for suitable non-free Schrodinger operators and Jacobi matrices.Comment: 11 page

Ergodic Jacobi matrices and conformal maps

We study structural properties of the Lyapunov exponent $\gamma$ and the density of states $k$ for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function $w=-\gamma+i\pi k$ as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem

Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.Comment: a preliminary version, 39 pages; some changes in the Introduction, Section 5 (Szeg\H o type asymptotics) is extende

Applications of Polynomial Chaos-Based Cokriging to Aerodynamic Design Optimization Benchmark Problems

In this work, the polynomial chaos-based Cokriging (PC-Cokriging) is applied to a benchmark aerodynamic design optimization problem. The aim is to perform fast design optimization using this multifidelity metamodel. Multifidelity metamodels use information at multiple levels of fidelity to make accurate and fast predictions. Higher amount of lower fidelity data can provide important information on the trends to a limited amount of high-fidelity (HF) data. The PC-Cokriging metamodel is a multivariate version of the polynomial chaos-based Kriging (PC-Kriging) metamodel and its construction is similar to Cokriging. It combines the advantages of the interpolation-based Kriging metamodel and the regression-based polynomial chaos expansions (PCE). In the work the PC-Cokriging model is compared to other metamodels namely PCE, Kriging, PC-Kriging and Cokriging. These metamodel are first compared in terms of global accuracy, measured by root mean squared error (RMSE) and normalized RMSE (NRMSE) for different sample sets, each with an increasing number of HF samples. These metamodels are then used to find the optimum. Once the optimum design is found computational fluid dynamics (CFD) simulations are rerun and the results are compared to each other. In this study a drag reduction of 73.1 counts was achieved. The multifidelity metamodels required 19 HF samples along with 1,055 low-fidelity to converge to the optimum drag value of 129 counts, while the single fidelity models required 155 HF samples to do the same

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials