A Study On The Cubature And The Least Deviation From Zero On Tetrahedron And Curved Tetrahedron

As we all know,in the univariate case,the first kind of Chebyshev polynomials are equivalent to cosine trigonometric functions.Many properties of Chebyshev polynomials,such as orthogonality,zero and extreme point properties and the least deviation from zero properties,can be obtained by studying cosine functions,and these properties are closely related to numerical integration and the best uniform approximation of functions,which is of great research significance.Many scholars such as sun Jiachang,Li Huiyuan and Xu Yuan have done a lot of research work on how to extend these results to multiple situations.In particular,in the document[1,2,3],using homogeneous coordinates and based on the study of the properties of generalized cosine function on multivariate simplex,gave the so-called Gauss-Lobatto type cubature formula on curved simplex.This paper mainly considers the construction of numerical integration on the tetrahedron and the curved tetrahedron,as well as the least deviation from zero on the curved tetrahedron.Firstly,this paper deeply studies the properties of generalized cosine function on tetrahedron,especially the extreme value properties of generalized cosine function TC3n,-n,-n,-n.Based on its extreme value points,an important discrete orthogonal property of generalized cosine function is proved,and a kind of cubature formula with trigonometric accuracy of degree n-1 on the curved tetrahedron domain is given.Then,the properties of the first kind of Generalized Chebyshev polynomials on the so-called curved tetrahedron equivalent to the generalized cosine function on the tetrahedron under the generalized cosine transformation are discussed.In particular,the extreme value properties of the Chebyshev polynomials Tn,0,0 are proved.From this,two kinds of alternating point groups of algebraic polynomial space on the curved tetrahedron are given,and the least deviation from zero of Tn,0,0 is proved.At the same time,a kind of cubature formula with algebraic accuracy of degree n-1 on the curved tetrahedron domain is also given.Compared with the cubature formula given in the literatures[2,3],which only provides odd algebraic accuracy,the cubature formulas given in this paper can provide algebraic accuracy of arbitrary degree,and for algebraic accuracy of degree 3 and 5,the number of nodes used in the formula given in this paper is the least.