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''Wilf formulas''

Quadrature formulas (cf. [[Quadrature formula|Quadrature formula]]) constructed from a [[Hilbert space|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|Bunyakovskii inequality]]) one has $|R[f]| \le \Vert R \Vert\cdot\Vert f \Vert$ 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 [[#References|[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 $b _ { \nu } = 0$ [[#References|[a1]]], [[#References|[a3]]].

The original construction of H.S. Wilf [[#References|[a4]]] was for the Hardy space (cf. also [[Hardy spaces|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|Gauss quadrature formula]]) [[#References|[a1]]]. They can be constructed from a suitable rational interpolant [[#References|[a1]]], [[#References|[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 $\int _ { a } ^ { b } ( f ^ { ( r ) } ( x ) ) ^ { 2 } d x \leq 1$ [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]] (see [[Optimal quadrature|Optimal quadrature]]; [[Best quadrature formula|Best quadrature formula]]). The Sard formula results from integrating the natural [[Spline|spline]] function of order $2 r - 1$ which interpolates $f$ at the nodes $x _ { 1 } , \ldots , x _ { n }$ [[#References|[a1]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  H. Braß,   "Quadraturverfahren" , Vandenhoeck&amp;Ruprecht  (1977)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  P.J. Davis,   P. Rabinowitz,   "Methods of numerical integration" , Acad. Press  (1984)  (Edition: Second)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  H. Engels,   "Numerical quadrature and cubature" , Acad. Press  (1980)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  H.S. Wilf,   "Exactness conditions in numerical quadrature"  ''Numer. Math.'' , '''6'''  (1964)  pp. 315–319</td></tr></table>