Chebyshev-type quadrature on multidimensional domains
Let \(E\) be a compact set in \(\mathbb{R}^ d\) with a finite positive measure \(\sigma\) on \(E\). A formula of the type \[ {1\over {\sigma(E)}} \int_ E f(x) d\sigma(x)= {1\over N} \sum_{j=1}^ N f(\xi_ j), \qquad \xi_ j\in E \tag{1.1} \] is called a Chebyshev type quadrature formula. It is said to be of degree \(p\) if the formula is exact for all polynomials of total degree \(\leq p\). The Gauss formula is exact of degree \(2m-1\) and is given by \[ {1\over \pi} \int_{-1}^ 1 f(x) {{dx} \over {\sqrt {1-x^ 2}}} = {1\over m} \sum_{j=1}^ m f(\xi_ j), \qquad \xi_ j= \cos {{2j-1} \over {2n}} \pi. \] The authors address the question of finding bounds on the number \(N= N(p)\) in (1.1) when there is a Chebyshev type formula (1.1) of degree \(p\). It is known from the work of Bernstein that if \(E=I= [-1,1]\), \(d \sigma(x)= dx\), then \(c_ 1 p^ 2\leq N\leq c_ 2 p^ 2\) (\(c_ 1\), \(c_ 2\) being constants) written as \(N\asymp p^ 2\). Here the authors show that in the case of the unit circle \(C\) with centre 0 and radius 1, \(N_ c(p)= p+1\). They prove the following estimates for ordinary surface area: (1) For the square \(Q= I^ 2\) in \(\mathbb{R}^ 2\), \(N_ Q(p) \asymp p^ 4\); (2) For the cylindrical surfaces \(CS= C\times I\) in \(\mathbb{R}^ 3\), \(N_{CS} (p) \asymp p^ 3\); (3) For the unit disc \(D\) in \(\mathbb{R}^ 2\), \(N_ D (p) \asymp p^ 3\); (4) For the unit shere \(S\) in \(\mathbb{R}^ 3\), \(c_ 1 p^ 2\leq N_ s (p)\leq c_ 2 p^ 3\). They obtain explicit constants in all cases and exhibit a Chebyshev formula in each case.
scientific article; zbMATH DE number 645063