奥田隆幸（東北大学）「Laplacian-type cubature formulas with smaller number of points」
This is a joint work with Masatake Hirao (Tokyo Woman's Christian University) and Masanori Sawa (Nagoya University).
It is known that each Euclidean t-design on the Euclidean space R^n can be considered as a Gaussian cubature formula of degree t on R^n. Fisher type inequalities for Euclidean even designs are also known as the Stroud bounds of Gaussian cubature formulas of even degree.
In this talk, we are interested in Laplacian-type cubature formulas as generalizations of Gaussian cubature formulas. Recall that in a Gaussian cubature of degree t, our fomula gives an equation between the integral of f and a positive weighted sum of f(x) of x in a finite set X for any polynomial f of degree at most t. That is, the integral operator is decomposed as a positive weighted sum of finite evaluation maps on the polynomial space of functions of degree at most t. In a Laplacian-type cubature fomula, we give a decomposition of the integral operator as a weighted sum of a Laplacian-type operator and finite evaluation maps.
The goal of this talk is to show generalizations of the Stroud bounds for Lapalacian-type cubature formulas on R^2 of even degree. We also give a sequence of examples of Lapalacian-type cubature formulas attained our lower bounds.