Abstract
The paper develops applications of symmetric orbit functions, known from irreducible representations of simple Lie groups, in numerical analysis. It is shown that these functions have remarkable properties which yield to cubature formulas, approximating a weighted integral of any function by a weighted finite sum of function values, in connection with any simple Lie group. The cubature formulas are specialized for simple Lie groups of rank two. An optimal approximation of any function by multivariate polynomials arising from symmetric orbit functions is discussed.
Highlights
The purpose of this paper is to extend the results of [1,2], where cubature formulas for numerical integration connected with three types of multivariate Chebyshev-like polynomials arising from Weyl group orbit functions are developed
The family of polynomials induced by the symmetric Weyl group orbit functions (C-functions) forms one of the most natural generalizations of the classical Chebyshev polynomials of one variable—the lowest symmetric orbit function arising from the Weyl group of A1 coincides with the common cosine function of one variable and induces the family of Chebyshev polynomials of the first kind [5]
Due to the generality of the present construction of the cubature formulas, some of the cases presented in this paper appeared already in the literature
Summary
The purpose of this paper is to extend the results of [1,2], where cubature formulas for numerical integration connected with three types of multivariate Chebyshev-like polynomials arising from Weyl group orbit functions are developed. The specific goal of this article is to derive the cubature rules and corresponding approximation methods for the family of the polynomials arising from symmetric exponential Weyl group orbit sums [3,4] and detail specializations of the general results for the two-variable polynomials. The simplex F together with the set of points FM have to be transformed via a transform which induces the corresponding family of polynomials (X-transform) This process results in the integration domain Ω of non-standard shape and the set of nodes ΩM , with distributed points inside Ω.
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