Multivariate Orthogonal Polynomials And Cubature Formulae

Orthogonal polynomials and numerical integration in several variables are two hot problems for discussion in the field of numerical approximation, and have many important applications in computational geometry, scientific computing, harmonic analysis, the theory of special functions, probability and statistics and so on. Furthermore both of them are also closely related. In this paper, some problems about these two topics are discussed, which mainly include some extensions of general properties of one dimensional orthogonal polynomials and the construction of cubature formulae with special structures.(1) Historically, the orthogonal polynomials originated in the theory of continued fractions. But continued fractions have been gradually abandoned as a starting point for the theory of orthogonal polynomials. One of the important reasons is that it is difficult to be extended. To overcome it, professor Luo Zhongxuan and professor Wang Renhong introduced the concepts of invariant factor for orthogonal polynomials of two variables. By invariant factor, they proved the Stieltjes theorem of two variables, in which invariant factors play an important role. In this paper, some further properties of invariant factor are discussed, especially the properties of its zeros. The following results are presented: Every zero of invariant factor must be an eigenvalue of the corresponding truncated Jacobi matrix, and hence real. Especially, if all the zeros of invariant factor are distinct from each other, then the invariant factor equals to the characteristic polynomial of the corresponding truncated Jacobi matrix up to a constant multiple. Moreover, for every zero of invariant factor, there must exist an orthogonal polynomial factored into two polynomials, from which an zero location of invariant factor, similar to the case of one dimensional orthogonal polynomial, can be obtained.(2) The properties of zeros for one dimensional orthogonal polynomial are the important part of the general properties. Usually to generalize the corresponding properties, common zeros of orthogonal polynomials are used in the case of several variables. Thus part of theory in one variable can be extended to several variables. But the common zeros location remains unknown. In one variable, all the zeros of orthogonal polynomials are located in the interior of integration interval, which is different from the case of two variables. The common zeros of all multivariate orthogonal polynomials with same total degree can lie outside of the region. An obvious example is the orthogonal polynomials on the two dimensional spherical shell. The original point is the common zero of all orthogonal polynomials with odd degree but lie outside of the spherical shell. Base on the properties of invariant factor we prove that all the common zeros lie in the interior of the convex closed hull of the integration region.(3) Numerical integration is closely related to orthogonal polynomials, and in one variable it usually takes the zeros of orthogonal polynomials as its knots which benefit from the good properties of univariate orthogonal polynomials. However, in higher dimension we know a litter about this. One of the difficulties is one does not know the exact number of common zeros for a set of polynomials. It follows from the property of invariant factor that for every zero of invariant factor there must exist an orthogonal polynomial which can be factored into a product of two polynomials of lower degree. Especially, for n=2, these two polynomials of lower degree is of degree 1. Stroud proved that if two orthogonal polynomials have four intersections, then these points must constitute a set of knots of a cubature formula of degree 3. Obviously, it is easy to compute the number of the common zeros of two orthogonal polynomials with the factoring property. It is based on this idea that we present a method to construct the cubature formula of degree 3 with 4 knots.(4) To investigate the properties of multivariate orthogonal polynomials, we extend the concept of invariant factor and get the similar results with two variables. The presented results include multivariate Stieltjes type theorem, the relations among invariant factor, multivariate orthogonal polynomials and Jacobi matrix, and common zeros location of multivariate orthogonal polynomials and so on.(5) Among various methods of constructing cubature formulae, the method of Cartesian product stands out with its simplicity. Briefly, the method for constructing product formulas is the method of separation of variables. If we can find a (nonlinear) transform which transforms the monomial integral into the product of d single integrals and if suitable formulas are known for these single integrals, then one can combine these formulas to give a formula. The most undesired property of product formulas is that the number of points increases very rapidly as d increases. To avoid it partly, we present a proper transformation which transforms the initial integration into the product of single integrations and reduces the number of knots greatly. Furthermore, some knots are used repeatedly by the cubature formulae of any degree.(6) Polynomial interpolation and quadrature formula are closely related. To investigate the interpolation problem of two variables, Liang presented a kind of recursive method of constructing properly posed set of nodes. This paper consider the corre- sponding recursive method of construction of numericai integration. First we present a construction method of cubature formulae with some knots along the selected algebraic curve. By selecting different algebraic curves and repeating the above process, we will obtain a cubature formula and then a recursive method. To reduce the number of the knots, we take the polynomial corresponding to the selected curve as an orthogonal polynomial. Thus the number of the knots of final cubature formula will be reduced greatly. Taking the unit disk as an example, a detailed construction process is given. Some cubature formula have gotten to the lower bound.(7) By the knowledge of polynomial ideal, we present a construction method of cubature formula with some fixed points. If we take the fixed points as the nodes of a cubature of lower degree and get a new cubature of higher degree, then an embedded formula sequence is obtained. This kind of embedded sequence can be used to estimate the errors. Furthermore, a noninterpolationary cubature formula is presented.