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]
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and
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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
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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
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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 |
Wilf quadrature formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wilf_quadrature_formulas&oldid=50278