The minimal number of nodes in Chebyshev type quadrature formulas

AbstractWe study Chebyshev type quadrature formulas of degree n with respect to a weight function on [-1,1], i.e. formulas 1∫−11w(t)dt∫−11f(t)w(t)dt=1N∑i=1Nf(xi)+R(f) with nodes xi∈[-1,1], such that R(f)=0 for every polynomial of degree ≤n. It is known that for a Jacobi weight function w(t)=(1−t)α(1+t)β the number of nodes has to satisfy the inequality N≥K1n2+2max(α,β) for some absolute constant K1>0. In this paper it is shown that for an ultra-spherical weight function w(t)=(1−t2)α with α≥0, this lower bound is of the right order, i.e. there exists a Chebyshev type quadrature formula of degree n with N≤K2n2+2α nodes. Our method of proof is based on a method of S.N. Bernstein who obtained the result in case α = 0. In general this method gives a large number of multiple nodes. It is also shown that the nodes can be chosen to be distinct