# 組合せ論セミナー

## 第51回 2014年2月18日 15:30〜17:00

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.