Absolutely continuous Jacobi operators

We show (among other results) that a symmetric Jacobi matrix whose diagonal is the zero sequence and whose super-diagonal 0$">satisfies , and has purely absolutely continuous spectrum when considered as a self-adjoint operator on .

     

Unbounded Jacobi matrices at critical coupling

We consider a class of Jacobi matrices with unbounded coefficients. This class is known to exhibit a first-order phase transition in the sense that, as a parameter is varied, one has purely discrete spectrum below the transition point and purely absolutely continuous spectrum above the transition point. We determine the spectral type and solution asymptotics at the transition point.     

A class of matrix-valued polynomials generalizing Jacobi polynomials

A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a three term recurrence relation, integral inter-relations, and weak orthogonality relations.     

Absolute Continuity in Periodic Waveguides

We study second order elliptic operators with periodic coefficientsin two-dimensional simply connected periodic waveguides withthe Dirichlet or Neumann boundary conditions. It is proved thatunder some mild smoothness restrictions on the coefficients,such operators have purely absolutely continuous spectra. Theproof follows a method suggested previously by A. Morame totackle periodic operators with variable coefficients in dimension2. 2000 Mathematical Subject Classification: 35J10, 35P05, 35J25.     

Asymptotics of Eigenvalues for a Class of Jacobi Matrices with Limiting Point Spectrum

In this paper, we study a class of Jacobi matrices with very rapidly decreasing weights. It is shown that the Weyl function (the matrix element of the resolvent of the operator) for the class under study can be expressed as the ratio of two entire transcendental functions of order zero. It is shown that the coefficients in the expansion of these functions in Taylor series are proportional to the generating functions of the number of integral solutions defined by certain Diophantine equations. An asymptotic estimate for the eigenvalues is obtained.     

Generalized Convolution for the Discrete Jacobi Transformation

We study the translation and the convolution associated to the discrete Jacobi transformation on complex sequences of slow and rapid growth. Also we establish new topological properties for these spaces of sequences. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Absolute Continuity of Gaussian Measures on Locally Compact Groups

This note discusses the topological necessary and sufficient conditions for a locally compact connected group to admit a Gaussian measure that is absolutely continuous with respect to Haar measure.     

Spectral Measures and Jacobi Matrices Related to Laguerre-Type Systems of Orthogonal Polynomials

This paper considers systems of Laguerre-type orthogonal polynomials for which the corresponding Jacobi matrices represent unbounded self-adjoint operators which are bounded above or below. Under appropriate assumptions on the coefficient sequences in the recursion formula, results are obtained on the uniform boundedness of the polynomials on bounded intervals, the absence of eigenvalues for the corresponding operator, and the absolute continuity of the measure of orthogonality. Date received: September 7, 1995. Date revised: April 17, 1996.     

On conjectured inequalities for zeros of Jacobi polynomials

Inequalities for the largest zero of Jacobi polynomials, conjectured recently by us and in joint work with P. Leopardi, are here extended to all zeros of Jacobi polynomials, and new relevant conjectures are formulated based on extensive computation.      

Conjectured inequalities for Jacobi polynomials and their largest zeros

Inequalities are conjectured for the Jacobi polynomials and their largest zeros. Special attention is given to the cases β = α − 1 and β = α.      

Anderson Localization for Jacobi Matrices Associated with High-Dimensional Skew Shifts

In this paper, the authors establish Anderson localization for a class of Jacobi matrices associated with skew shifts on T^d, d ≥ 3.     

An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations

This paper aims to construct a general formulation for the Jacobi operational matrix of fractional integral operator. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the Jacobi integral operational matrix to the fractional calculus. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Continuity of the Filter with Unbounded Observation Coefficients

In this article, we study the continuity with respect to the trajectories of the observation process for the filter associated with nonlinear filtering problems when the coefficients depend on both the signal and the observation and the observation coefficient is unbounded.

To achieve this task we define a formal unnormalized filter and prove by limiting arguments that it is related to the original filter through a generalized Bayes formula, and is locally Lipschitz continuous with respect to the uniform norm.     

A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients

This paper presents a shifted fractional‐order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time‐fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.     

Inversion of Vandermonde-Like Matrices

Let {x }_=0,...,n be a set of distinct nodes, and let {p _n } _n0 be a sequence of polynomials, degp _n =n, whose elements satisfy a three-term recurrence relation. We derive explicit representations for the inverse of a Vandermonde-like matrix V _n =(p (x ))_,=0,...,n and present some numerical results.     

An extremal property of Jacobi polynomials in two-sided Chernoff-type inequalities for higher order derivatives

For a weight function generating the classical Jacobi polynomials, the sharp double estimate of the distance from the subspace of all polynomials of an arbitrary fixed order is established.

     

The higher-order differential operator for the generalized Jacobi polynomials — new representation and symmetry

For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by Koornwinder’s generalized Jacobi polynomials with four parameters $\alpha ,\beta \in {\mathbb{N}}_0$ and $M,N\ge 0$ determining the orthogonality measure on the interval $?1\le x\le 1$. The corresponding differential equation of order $2\alpha +2\beta +6$ is presented here as a linear combination of four elementary components which make the corresponding differential operator widely accessible for applications. In particular, we show that this operator is symmetric with respect to the underlying scalar product and thus verify the orthogonality of the eigenfunctions.