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1.
In this work, semiclassical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated with a quasi definite linear functional satisfying a matrix Pearson-type differential equation. Semiclassical functionals are characterized by means of the analogue of the structure relation in one variable. Moreover, non trivial examples of semiclassical orthogonal polynomials in two variables are given.  相似文献   

2.
In this work, we introduce the classical orthogonal polynomials in two variables as the solutions of a matrix second order partial differential equation involving matrix polynomial coefficients, the usual gradient operator, and the divergence operator. Here we show that the successive gradients of these polynomials also satisfy a matrix second order partial differential equation closely related to the first one.  相似文献   

3.
In the first part we expose the notion of continued fractions in the matrix case. In this paper we are interested in their connection with matrix orthogonal polynomials.

In the second part matrix continued fractions are used to develop the notion of matrix Chebyshev polynomials. In the case of hermitian coefficients in the recurrence formula, we give the explicit formula for the Stieltjes transform, the support of the orthogonality measure and its density. As a corollary we get the extension of the matrix version of the Blumenthal theorem proved in [J. Approx. Theory 84 (1) (1996) 96].

The third part contains examples of matrix orthogonal polynomials.  相似文献   


4.
In this paper, we consider bivariate orthogonal polynomials associated with a quasi-definite moment functional which satisfies a Pearson-type partial differential equation. For these polynomials differential properties are obtained. In particular, we deduce some structure and orthogonality relations for the successive partial derivatives of the polynomials.   相似文献   

5.
Classical orthogonal polynomials in two variables can be characterized as the polynomial solutions of a matrix second-order partial differential equation involving matrix polynomial coefficients. In this work, we study classical orthogonal polynomials in two variables whose partial derivatives satisfy again a second-order partial differential equation of the same type.  相似文献   

6.
We state and prove characterization theorem for semi-classical orthogonal polynomials on nonuniform lattices (quadratic lattices of a discrete or q-discrete variable). This theorem proves the equivalence between the four characterization properties, namely, the Pearson type equation for the linear functional, the strictly quasi-orthogonality of the derivatives, the structure relation, and the Riccati equation for the formal Stieltjes function. We give the classification of the semi-classical linear functional of class one on nonuniform lattice. Using the definition and the properties of the associated orthogonal polynomials, we prove that semi-classical orthogonal polynomials satisfy the second-order divided difference equation on nonuniform lattices.  相似文献   

7.
For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix is Hankel and positive definite. The polynomials satisfy a three term recurrence relation. When the measure is supported on the complex unit circle, the moment matrix is positive definite and Toeplitz. Then they satisfy a coupled Szeg recurrence relation but also a three term recurrence relation. In this paper we study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. The relation with Padé approximation and with Krylov subspace iterative methods is also outlined.This research was supported by the National Fund for Scientific Research (NFWO), project Lanczos, grant #2.0042.93.  相似文献   

8.
This paper is devoted to the study of direct and inverse (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree.The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non-necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters.The non-uniqueness of the inverse problem makes it more interesting than the direct one. Due to this reason, special attention is paid to the inverse modification, showing that different approaches are possible depending on the data about the polynomial modification at hand. These different approaches are translated as different kinds of initial conditions for the related inverse algorithm.Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a certain type of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.  相似文献   

9.
Some families of orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n have recently been introduced (see [Internat. Math. Res. Notices 10 (2004) 461–484]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues’ formulas of the type (ΦnW)(n)W-1, where Φ is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues’ formula, well suited to the matrix case, appears in [Internat. Math. Res. Notices 10 (2004) 482].In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues’ formula and show that scalar type Rodrigues’ formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar-type Pearson equation as well as that of a noncommutative version of it.  相似文献   

10.
Differential properties for orthogonal polynomials in several variables are studied. We consider multivariate orthogonal polynomials whose gradients satisfy some quasi-orthogonality conditions. We obtain several characterizations for these polynomials including the analogues of the semiclassical Pearson differential equation, the structure relation and a differential-difference equation.  相似文献   

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