Theory Of Orthogonal Polynomials Research Articles

Perturbations of Christoffel–Darboux Kernels: Detection of Outliers

Two central objects in constructive approximation, the Christoffel–Darboux kernel and the Christoffel function, encode ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel–Darboux kernel in $$\mathbb {C}^d$$ , with emphasis on the perturbation of Christoffel functions and their level sets with respect to perturbations of small norm or low rank. The statistical notion of leverage score provides a quantitative criterion for the detection of outliers in large data. Using the refined theory of Bergman orthogonal polynomials, we illustrate the main results, including some numerical simulations, in the case of finite atomic perturbations of area measure of a 2D region. Methods of function theory of a complex variable and (pluri) potential theory are widely used in the derivation of our perturbation formulas.

A matrix approach to some second-order difference equations with sign-alternating coefficients

ABSTRACTIn this paper, we analyse and unify some recent results on the double sequence , for , defined by the second-order difference equation with and , in terms of matrix theory and orthogonal polynomials theory. Moreover, we provide a general solution to using a closely related approach. We discuss briefly other recent problems involving a general recurrence relation of second order and relate them with the existing literature.

Finite term relations for the exponential orthogonal polynomials

The exponential orthogonal polynomials encode via the theory of hyponormal operators a shade functiongsupported by a bounded planar shape. We prove under natural regularity assumptions that these complex polynomials satisfy a three term relation if and only if the underlying shape is an ellipse carrying uniform black on white. More generally, we show that a finite term relation among these orthogonal polynomials holds if and only if the first row in the associated Hessenberg matrix has finite support. This rigidity phenomenon is in sharp contrast with the theory of classical complex orthogonal polynomials. On function theory side, we offer an effective way based on the Cauchy transforms ofg,z̅g,…,z̅dg, to decide whether a (d+ 2)-term relation among the exponential orthogonal polynomials exists; in that case we indicate how the shade functiongcan be reconstructed from a resulting polynomial of degreedand the Cauchy transform ofg. A discussion of the relevance of the main concepts in Hele-Shaw dynamics completes the article.

PolyChaos.jl — A Julia Package for Polynomial Chaos in Systems and Control

Polynomial chaos expansion (pce) is an increasingly popular technique for uncertainty propagation and quantification in systems and control. Based on the theory of Hilbert spaces and orthogonal polynomials, PCE allows for a unifying mathematical framework to study systems under arbitrary uncertainties of finite variance; we introduce this problem as a so-called mapping under uncertainty. For practical PCE-based applications we require orthogonal polynomials relative to given probability densities, and their quadrature rules. With PolyChaos.jl we provide a Julia software package that delivers the desired functionality: given a probability density function, PolyChaos.jl offers several numerical routines to construct the respective orthogonal polynomials, and the quadrature rules together with tensorized scalar products. PolyChaos.jl is the first PCE-related software written in Julia, a scientific programming language that combines the readability of scripted languages with the speed of compiled languages. We provide illustrating numerical examples that show both PCE and PolyChaos.jl in action.

Representation of the quantum mechanical wavefunction by orthogonal polynomials in the energy and physical parameters

We present a formulation of quantum mechanics based on the theory of orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable quantum mechanical problems becomes larger than in the conventional formulation (see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases (the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.

Tridiagonal representation approach in quantum mechanics

We present an algebraic approach for finding exact solutions of the wave equation. The approach, which is referred to as the tridiagonal representation approach, is inspired by the J-matrix method and based on the theory of orthogonal polynomials. The class of exactly solvable problems in this approach is larger than the conventional class. All properties of the physical system (energy spectrum of the bound states, phase shift of the scattering states, energy density of states, etc) are obtained in this approach directly and simply from the properties of the associated orthogonal polynomials.

Matrix factorizations and orthogonal polynomials

We present some elements of the theory of orthogonal polynomials based on matrix decompositions. We focus our attention on discrete linear functionals, and use the Meixner polynomials as a concrete example.

On conjectures of Stenger in the theory of orthogonal polynomials

The conjectures in the title deal with the zeros x_{j}, j=1,2, ldots ,n, of an orthogonal polynomial of degree n>1 relative to a nonnegative weight function w on an interval [a,b] and with the respective elementary Lagrange interpolation polynomials ell _{k} ^{(n)} of degree n-1 taking on the value 1 at the zero x_{k} and the value 0 at all the other zeros x_{j}. They involve matrices of order n whose elements are integrals of ell _{k}^{(n)}, either over the interval [a,x_{j}] or the interval [x_{j},b], possibly containing w as a weight function. The claim is that all eigenvalues of these matrices lie in the open right half of the complex plane. This is proven to be true for Legendre polynomials and a special Jacobi polynomial. Ample evidence for the validity of the claim is provided for a variety of other classical, and nonclassical, weight functions when the integrals are weighted, but not necessarily otherwise. Even in the case of weighted integrals, however, the conjecture is found by computation to be false for a piecewise constant positive weight function. Connections are mentioned with the theory of collocation Runge–Kutta methods in ordinary differential equations.

Poncelet's theorem, paraorthogonal polynomials and the numerical range of compressed multiplication operators

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.

Geometry of curves in [formula omitted] from the local singular value decomposition

We establish a connection between the local singular value decomposition and the geometry of n-dimensional curves. In particular, we link the left singular vectors to the Frenet-Serret frame, and the generalized curvatures to the singular values. Specifically, let γ:I→Rn be a parametric curve of class Cn+1, regular of order n. The Frenet-Serret apparatus of γ at γ(t) consists of a frame e1(t),…,en(t) and generalized curvature values κ1(t),…,κn−1(t). Associated with each point of γ there are also local singular vectors u1(t),…,un(t) and local singular values σ1(t),…,σn(t). This local information is obtained by considering a limit, as ϵ goes to zero, of covariance matrices defined along γ within an ϵ-ball centered at γ(t). We prove that for each t∈I, the Frenet-Serret frame and the local singular vectors agree at γ(t) and that the values of the curvature functions at t can be expressed as a fixed multiple of a ratio of local singular values at t. To establish this result we prove a general formula for the recursion relation of a certain class of sequences of Hankel determinants using the theory of monic orthogonal polynomials and moment sequences.

Towards an Algebraic Theory of Orthogonal Polynomials in Several Variables

In this paper, we review on a general theory of orthogonal polynomials in several variables (O.P.S.V) in which we present two different approaches for the three-term recurrence relation. We draw attention to the fact that it is possible to take advantage of the orthogonal projection approach of the three-term recurrence relation towards the development of the algebraic theory of O.P.S.V.

On Modified Multi-Output Chebyshev-Polynomial Feed-Forward Neural Network for Pattern Classification of Wine Regions

Wine, an alcoholic beverage made from fermented grapes, has become an increasingly popular drink. However, wine regions may directly affect the quality and taste of the wine, and misjudgment of the wine regions leads to confusion for dealers and consumers in choosing wine types. In recent decades, different methods in machine learning have been presented and investigated the pattern classification. In this paper, based on the existing results, a modified multi-output Chebyshev-polynomial feed-forward neural network (MOCPFFNN) is presented, analyzed, and applied to the pattern classification of wines regions. According to the orthogonal polynomial theory, the activation functions of the MOCPFFNN are improved to some Chebyshev polynomials. In addition, to have a lower computational complexity, the presented neural network model is automatically determined by the eight-fold cross validation (8FCV) algorithm and the weight direct determination (WDD) algorithm. Finally, comparisons are made among the presented model and other classical methods, e.g., feed-forward back propagation (FFBP), layer recurrent neural network (LRNN), and nonlinear auto regressive with exogenous inputs (NARX), K-nearest neighbors (KNN), random forest, which confirm that the modified MOCPFFNN has the best approximation and generalization performance in the pattern classification of wine regions, with the accuracy rates of the training set and test set reaching 99.17% and 94.83%, respectively. Moreover, the variance of the accuracy of the presented MOCPFFNN method in the experiments is 0, which illustrates its high robustness in pattern classification.

Quantum Systems Associated with the Hahn and Continuous Hahn Polynomials

Using a formulation of quantum mechanics based on the theory of orthogonal polynomials, we introduce a four-parameter system associated with the Hahn and continuous Hahn polynomials. The continuum energy scattering states are written in terms of the continuous Hahn polynomial whose asymptotics give the scattering amplitude and phase shift. On the other hand, the finite number of discrete bound states are associated with the Hahn polynomial.

Spectral collocation methods for nonlinear weakly singular Volterra integro‐differential equations

In this paper, a spectral collocation approximation is proposed for neutral and nonlinear weakly singular Volterra integro‐differential equations (VIDEs) with non‐smooth solutions. We use some suitable variable transformations to change the original equation into a new equation, so that the solution of the resulting equation possesses better regularity, and the the Jacobi orthogonal polynomial theory can be applied conveniently. Under reasonable assumptions on the nonlinearity, we carry out a rigorous error analysis in L∞ norm and weighted L2 norm. To perform the numerical simulations, some test examples (linear and nonlinear) are considered with nonsmooth solutions, and numerical results are presented. Further more, the comparative study of the proposed methods with some existing numerical methods is provided.

Matricial Baxter's theorem with a Nehari sequence

AbstractIn the theory of orthogonal polynomials, (non‐trivial) probability measures on the unit circle are parametrized by the Verblunsky coefficients. Baxter's theorem asserts that such a measure is absolutely continuous and has positive density with summable Fourier coefficients if and only if its Verblusnky coefficients are summable. This note presents a version of Baxter's theorem in the matrix case from a viewpoint of the Nehari problem.

On moving frames and Toda lattices of BKP and CKP types

This paper is mainly concerned with the geometric formulations of Toda lattices of BKP and CKP types as evolutions of invariants and their explicit expressions. The theory for discrete curves in centro-affine geometry is constructed by using the method of a moving frame. With the help of orthogonal polynomial theory, we choose the appropriate evolutions for the curves and the evolutions induced on invariants are related to these Toda-type lattices. In addition, the explicit expressions for the discrete invariant curve flows, discrete moving frames and Maurer–Cartan invariants are provided.

Acoustic imaging of layered media

This paper presents the echoes-to-impedance transform, a nonlinear transform designed for acoustic imaging of layered media—for example, sedimentary geological formations, biological tissue such as skin, or laminated structures in the built environment. The transform converts time domain digital reflection data directly into impedance as a function of spatial location, using minimal prior information about the source wavelet and no prepreprocessing. It is simple, fast, and, according to numerical experiments, highly accurate. More than this, physical structure is superresolved at a finer scale than that of the source wavelet. The derivation of the echoes-to-impedance transform stems from a recently developed numerical method for wave propagation in one dimension in conjunction with the theory of orthogonal polynomials on the unit circle.

On sequences of Hurwitz polynomials related to orthogonal polynomials

ABSTRACTIn this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.

Gibbs–Wilbraham oscillation related to an Hermite interpolation problem on the unit circle

The aim of this piece of work is to study some topics related to an Hermite interpolation problem on the unit circle. We consider as nodal points the zeros of the para-orthogonal polynomials with respect to a measure in the Baxter class and such that the sequence of the first derivative of the reciprocal of the orthogonal polynomials is uniformly bounded on the unit circle. We study the convergence of the Hermite–Fejér interpolants related to piecewise continuous functions and we describe the sets in which the interpolants uniformly converge to the piecewise continuous function as well as the oscillatory behavior of the interpolants near the discontinuities, where a Gibbs–Wilbraham phenomenon appears. Finally we present some numerical experiments applying the main results and by considering nodal systems of interest in the theory of orthogonal polynomials.

The Cauchy Two-Matrix Model, C-Toda Lattice and CKP Hierarchy

This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hi- erarchy with the help of orthogonal polynomials theory and Toda-type equations. Starting from the symmetric reduction of Cauchy biorthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing time flows. Then, matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy which reveals the time-dependent partition function of the Cauchy two-matrix model is nothing but the {\tau}-function of the CKP hiearchy. At last, the connection between the Cauchy two-matrix model and Bures ensemble is established from the point of view of integrable systems.