Optimal quadratures for analytic functions

Abstract

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 ^o _i and weightsA ^o _i , for a fixedn, which minimize the errorE _n (f)=∫ ¹₋₁ w(x)f(x)dx − Σ ⁼¹ⁿ _i 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 ^G _i , 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 ^G _i . In each case,A ^G _i =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 ^G _i ,A _i *;i=1,...,n} to provides the optimal abscissas and weights. For otherw(x), we study the closeness of the set {x ^G _i ,A _i *;i=1,...,n} to the optimal solution {x ^o _i ,A ^o _i ;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.