Wilf quadrature formulas

Wilf formulas

Quadrature formulas (cf. Quadrature formula) constructed from a Hilbert space setting.

Let be a Hilbert space of continuous functions such that and are continuous functionals; let for . Riesz's representation theorem guarantees the existence of an such that . By the Schwarz inequality (cf. Bunyakovskii inequality) one has in the Hilbert space norm. The formula is called optimal in if and are chosen such as to minimize . If has a continuously differentiable reproducing kernel , then such optimal formulas necessarily satisfy [a1]

and

and () if (). Here, denotes the derivative with respect to the first variable. Formulas which satisfy these conditions are called Wilf formulas.

The problem of minimizing can also be considered for fixed nodes . These formulas are characterized by integrating the unique element of least norm in which interpolates at the nodes . An analogous statement holds for Hermite quadrature formulas of the type . The Wilf formula for free nodes is the Wilf formula for those fixed nodes for which [a1], [a3].

The original construction of H.S. Wilf [a4] was for the Hardy space (cf. also Hardy spaces) of functions which are analytic inside the open disc with radius , with inner product

In the Hardy space the necessary conditions have a unique solution. The nodes are in , the weights are positive and . For fixed and these formulas converge to the Gaussian formulas (cf. also Gauss quadrature formula) [a1]. They can be constructed from a suitable rational interpolant [a1], [a3].

For fixed nodes , the inner product

leads to the Sard quadrature formula, which is optimal in the class of functions with [a1], [a2], [a3] (see Optimal quadrature; Best quadrature formula). The Sard formula results from integrating the natural spline function of order which interpolates at the nodes [a1].

References

[a1]	H. Braß, "Quadraturverfahren" , Vandenhoeck&Ruprecht (1977)
[a2]	P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984) (Edition: Second)
[a3]	H. Engels, "Numerical quadrature and cubature" , Acad. Press (1980)
[a4]	H.S. Wilf, "Exactness conditions in numerical quadrature" Numer. Math. , 6 (1964) pp. 315–319