Anti-Gaussian quadrature formulae of Chebyshev type

Abstract

We prove that there is no positive measure d σ d\sigma on the interval [ a , b ] [a,b] such that the corresponding anti-Gaussian quadrature formula is also a Chebyshev quadrature formula. We also show that the only positive and even measure d σ ( t ) = d σ ( − t ) d\sigma (t)=d\sigma (-t) on the symmetric interval [ − a , a ] [-a,a] , for which the anti-Gaussian formula has the form ∫ − a a f ( t ) d σ ( t ) = μ 0 2 [ f ( a ) + f ( − a ) ] + R 2 A G ( f ) \int _{-a}^{a}f(t)d\sigma (t)=\frac {\mu _{0}}{2}[f(a)+f(-a)]+R_{2}^{AG}(f) for n = 1 n=1 and ∫ − a a f ( t ) d σ ( t ) = w 1 f ( a ) + w ∑ μ = 2 n f ( t μ ) + w 1 f ( − a ) + R n + 1 A G ( f ) \int _{-a}^{a}f(t)d\sigma (t)=w_{1}f(a)+w\sum _{\mu =2}^{n}f(t_{\mu })+w_{1}f(-a)+R_{n+1}^{AG}(f) for all n ≥ 2 n\geq 2 , is the measure d σ ( t ) = ( a 2 − t 2 ) − 1 / 2 d t d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt . It turns out that the formula for n ≥ 2 n\geq 2 is the ( n − 1 ) (n-1) -point Gauss-Lobatto quadrature formula for the measure d σ ( t ) = ( a 2 − t 2 ) − 1 / 2 d t d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt , which is a generalization of what happens in the case of the Chebyshev measure of the first kind. Moreover, we compute the anti-Gaussian formulae for any one of the four Chebyshev measures.