Wilf quadrature formulas

Wilf formulas

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

Let $\mathcal{H}$ be a Hilbert space of continuous functions such that $I [ f ] = \int _ { a } ^ { b } f ( x ) d x$ and $L _ { \nu } [ f ] = f ( x _ { \nu } )$ are continuous functionals; let $R = I - \sum _ { \nu = 1 } ^ { n } \alpha _ { \nu } L _ { \nu }$ for $( \alpha _ { 1 } , \dots , \alpha _ { n } ) \in \mathbf{C} ^ { n }$. Riesz's representation theorem guarantees the existence of an $r \in \mathcal{H}$ such that $R [ f ] = ( r , f )$. By the Schwarz inequality (cf. Bunyakovskii inequality) one has in the Hilbert space norm. The formula is called optimal in $\mathcal{H}$ if $x _ { 1 } , \ldots , x _ { n }$ and $\alpha_{1} , \ldots , \alpha _ { n }$ are chosen such as to minimize $\| r\|$. If $\mathcal{H}$ has a continuously differentiable reproducing kernel $K$, then such optimal formulas necessarily satisfy [a1]

\begin{equation*} R [ K ( x _ { \nu } , . ) ] = 0 , \quad \nu = 1 , \dots , n, \end{equation*}

and

\begin{equation*} R [ K _ { x } ( x _ { \nu } , . ) ] = 0 , \quad \nu = 2 , \dots , n - 1, \end{equation*}

and $\nu = 1$ ($\nu = n$) if $x _ { 1 } \neq a$ ($x _ { n } \neq b$). Here, $K _ { x }$ denotes the derivative with respect to the first variable. Formulas which satisfy these conditions are called Wilf formulas.

The problem of minimizing $\| r\|$ can also be considered for fixed nodes $x _ { 1 } , \ldots , x _ { n }$. These formulas are characterized by integrating the unique element of least norm in $\mathcal{H}$ which interpolates $f$ at the nodes $x _ { \nu }$. An analogous statement holds for Hermite quadrature formulas of the type $\sum _ { \nu = 1 } ^ { n } \alpha _ { \nu } f ( x _ { \nu } ) + \sum _ { \nu = 1 } ^ { n } \beta _ { \nu } f ^ { \prime } ( x _ { \nu } )$. The Wilf formula for free nodes is the Wilf formula for those fixed nodes for which $b _ { \nu } = 0$ [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 $\rho$, with inner product

\begin{equation*} ( f , g ) = \operatorname { lim } _ { \eta \rightarrow \rho - 0 } \int _ { | z | = \eta } f ( z ) \overline { g ( z ) } d s. \end{equation*}

In the Hardy space the necessary conditions have a unique solution. The nodes are in $[ - 1,1 ]$, the weights are positive and $\sum _ { \nu = 1 } ^ { n } \alpha _ { \nu } \leq 2$. For fixed $n$ and $\rho \rightarrow \infty$ 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 $x _ { 1 } , \ldots , x _ { n }$, the inner product

\begin{equation*} ( f , g ) = \sum _ { \nu = 1 } ^ { r } f ( x _ { \nu } ) g ( x _ { \nu } ) + \int _ { a } ^ { b } f ^ { ( r ) } ( x ) g ^ { ( r ) } ( x ) d x \end{equation*}

leads to the Sard quadrature formula, which is optimal in the class of functions $f$ with $\int _ { a } ^ { b } ( f ^ { ( r ) } ( x ) ) ^ { 2 } d x \leq 1$ [a1], [a2], [a3] (see Optimal quadrature; Best quadrature formula). The Sard formula results from integrating the natural spline function of order $2 r - 1$ which interpolates $f$ at the nodes $x _ { 1 } , \ldots , x _ { n }$ [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