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''Wilf formulas''
 
''Wilf formulas''
  
 
Quadrature formulas (cf. [[Quadrature formula|Quadrature formula]]) constructed from a [[Hilbert space|Hilbert space]] setting.
 
Quadrature formulas (cf. [[Quadrature formula|Quadrature formula]]) constructed from a [[Hilbert space|Hilbert space]] setting.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w1202001.png" /> be a Hilbert space of continuous functions such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w1202002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w1202003.png" /> are continuous functionals; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w1202004.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w1202005.png" />. Riesz's representation theorem guarantees the existence of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w1202006.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w1202007.png" />. By the Schwarz inequality (cf. [[Bunyakovskii inequality|Bunyakovskii inequality]]) one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w1202008.png" /> in the Hilbert space norm. The formula is called optimal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w1202009.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020011.png" /> are chosen such as to minimize <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020013.png" /> has a continuously differentiable reproducing kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020014.png" />, then such optimal formulas necessarily satisfy [[#References|[a1]]]
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w1202008.png"/> 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]]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020015.png" /></td> </tr></table>
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\begin{equation*} R [ K ( x _ { \nu } , . ) ] = 0 , \quad \nu = 1 , \dots , n, \end{equation*}
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020016.png" /></td> </tr></table>
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\begin{equation*} R [ K _ { x } ( x _ { \nu } , . ) ] = 0 , \quad \nu = 2 , \dots , n - 1, \end{equation*}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020017.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020018.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020019.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020020.png" />). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020021.png" /> denotes the derivative with respect to the first variable. Formulas which satisfy these conditions are called Wilf formulas.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020022.png" /> can also be considered for fixed nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020023.png" />. These formulas are characterized by integrating the unique element of least norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020024.png" /> which interpolates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020025.png" /> at the nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020026.png" />. An analogous statement holds for Hermite quadrature formulas of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020027.png" />. The Wilf formula for free nodes is the Wilf formula for those fixed nodes for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020028.png" /> [[#References|[a1]]], [[#References|[a3]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020029.png" />, with inner product
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020030.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020031.png" />, the weights are positive and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020032.png" />. For fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020034.png" /> 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]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020035.png" />, the inner product
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For fixed nodes $x _ { 1 } , \ldots , x _ { n }$, the inner product
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020036.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020037.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020038.png" /> [[#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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020039.png" /> which interpolates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020040.png" /> at the nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120200/w12020041.png" /> [[#References|[a1]]].
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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====
 
====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>
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<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>

Revision as of 16:58, 1 July 2020

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
How to Cite This Entry:
Wilf quadrature formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wilf_quadrature_formulas&oldid=50278
This article was adapted from an original article by Sven Ehrich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article