Optimal quadratures for analytic functions

For integrals∫ −1 1 w(x)f(x)dx with $$w(x) = (1 - x)^{ \pm \tfrac{1}{2}} (1 + x)^{ \pm \tfrac{1}{2}} $$ and with analytic integrands, we consider the determination of “optimal” abscissasx and weightsA , for a fixedn, which minimize the errorE n (f)=∫ −1 1 w(x)f(x)dx − Σ =1 A i f(x i ) over an appropriate Hilbert spaceH 2(E ρ ; ∣w(z)∣) of analytic functions. Simultaneously, we consider the simpler problem of determining “intermediate-optimal” weightsA i *, corresponding to (preassigned) Gaussian abscissasx , which minimize the quadrature error. For eachw(x), the intermediate-optimal weightsA i * are obtained explicitly, and these come out proportional to the corresponding Gaussian weightsA . In each case,A =A i *+O(ϱ −4n ),ϱ → ∞. For $$w(x) = (1 - x^2 )^{ \pm \tfrac{1}{2}} $$ , a complete explicit solution for optimal abscissas and weights is given; in fact, the set {x ,A i *;i=1,...,n} to provides the optimal abscissas and weights. For otherw(x), we study the closeness of the set {x ,A i *;i=1,...,n} to the optimal solution {x ,A ;i=1,...,n} in terms ofε n (ϱ), the maximum absolute remainder in the second set ofn normal equations. In each case,ε n (ϱ) is, at least, of the order ofϱ −4n for largeϱ.