Note on the Algebraic Precision of Reducing-Dimensionality Expansion

In formula (1) we can choose an auxilliary polynomial P(x) from the class of polyncmials K_m={P_m(x)|P_m(x)∈P_m and the coefficient of the term x_1~(m_3)x_2~(m_2)…x_n~(m_n) is 1, m_1+m_2+…+m_n=m}, in which P_m denotes the space of all polynomials of degree≤m. Theorem For any bounded region in R~n and any given positive integer m, there exists a class of auxilliary polynomials P(x)∈K_m in (1) such that the reducingdimensionality expansion (1) is of the highest algebraic precision 2m-1.     

一类轴对称平面域上的求积公式

§1 引 言 设二维区域Ω,权函数p(x,y)0,(x,y)∈Ω。寻求以下的求积公式 y≈sum from j=1 to N(c_j(x_j,y_j)), (1.1)使其具有m次代数精度而结点数N为最小,其中c_j为权系数,(x_j,y_j)为结点,j=1,2,…N。我们称具有这种性质的求积公式为具有m次代数精度的最少结点求积公式,简称为最少结点求积公式。 研究各种求积公式中结点数下界,以及构造出各种区域上最少结点求积公式是很有意义的问题。由于求积公式的结点数下界对于固定的代数精度而言,是随积分区域而变化的。因此,只能对各种具体的区域来研究结点数下界的问题。例如和H.Moller