Stieltjes polynomials and Gauss-Kronrod quadrature formulae for measures induced by Chebyshev polynomials

Abstract

Given a fixedn≥1, and a (monic) orthogonal polynomial π _n (·)=π _n (·;dσ) relative to a positive measuredσ on the interval [a, b], one can define the nonnegative measure$d\hat \sigma _n (t) = [\pi _n (t;d\sigma )]^2 d\sigma (t)$, to which correspond the (monic) orthogonal polynomials$\hat \pi _{m,n} ( \cdot ) = \pi _m ( \cdot ;d\hat \sigma _n ),m = 0,1,2,...$. The coefficients in the three-term recurrence relation for$\hat \pi _{m,n}$, whendσ is a Chebyshev measure of any of the four kinds, were obtained analytically in closed form by Gautschi and Li. Here, we give explicit formulae for the Stieltjes polynomials$\hat \pi _{n + 1,n}^ * ( \cdot ) = \pi _{n + 1}^ * ( \cdot ;d\hat \sigma _n )$ whendσ is any of the four Chebyshev measures. In addition, we show that the corresponding Gauss-Kronrod quadrature formulae for each of these$d\hat \sigma _n$, based on the zeros of$\hat \pi _{n,n}$ and$\hat \pi _{n + 1,n}^ *$, have all the desirable properties of the interlacing of nodes, their inclusion in [−1, 1], and the positivity of all quadrature weights. Exceptions occur only for the Chebyshev measuredσ of the third or fourth kind andn even, in which case the inclusion property fails. The precise degree of exactness for each of these formulae is also determined.