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An [[Interpolation process|interpolation process]] constructed from a given interpolation process by imposing additional interpolation conditions.
 
An [[Interpolation process|interpolation process]] constructed from a given interpolation process by imposing additional interpolation conditions.
  
 
==The method of additional nodes.==
 
==The method of additional nodes.==
The method of additional nodes was introduced by J. Szabados in [[#References|[a4]]] to approximate the derivatives of a function by means of the derivative of the Lagrange interpolating polynomial (cf. also [[Lagrange interpolation formula|Lagrange interpolation formula]]). For given interpolation nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e1202701.png" />, the method consists in considering the interpolation process with respect to the nodes
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The method of additional nodes was introduced by J. Szabados in [[#References|[a4]]] to approximate the derivatives of a function by means of the derivative of the Lagrange interpolating polynomial (cf. also [[Lagrange interpolation formula]]). For given interpolation nodes $x _ { 1 } < \ldots < x _ { m }$, the method consists in considering the interpolation process with respect to the nodes
  
<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/e/e120/e120270/e1202702.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e1202702.png"/></td> </tr></table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e1202703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e1202704.png" /> are equidistant nodes between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e1202705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e1202706.png" />, respectively between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e1202707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e1202708.png" />.
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where $y _ { i }$, $z_i$ are equidistant nodes between $a$ and $x_{1} $, respectively between $x _ { m }$ and $b$.
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e1202709.png" /> being the zeros of the Jacobi polynomial (cf. [[Jacobi polynomials|Jacobi polynomials]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027011.png" />, the [[Lebesgue constants|Lebesgue constants]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027012.png" /> of the Lagrange interpolating polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027013.png" /> have the behaviour
+
For $x _ { 1 } , \ldots , x _ { m }$ being the zeros of the Jacobi polynomial (cf. [[Jacobi polynomials|Jacobi polynomials]]) $P _ { m } ^{( \alpha , \beta )}$, $\alpha , \beta &gt; - 1$, the [[Lebesgue constants|Lebesgue constants]] $\Lambda _ { m } ^ { \alpha , \beta }$ of the Lagrange interpolating polynomials $p _ { m } ^ { \alpha , \beta }$ have the behaviour
  
<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/e/e120/e120270/e12027014.png" /></td> </tr></table>
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\begin{equation*} \Lambda _ { m } ^ { \alpha , \beta } \sim \operatorname { max } \{ \operatorname { log } m , m ^ { \gamma + 1 / 2 } \}, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027015.png" />. Therefore, only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027016.png" /> will <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027017.png" /> have the optimal behaviour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027018.png" />. Denoting by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027019.png" /> the Lebesgue constant of the extended interpolation process, one has
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where $\gamma = \operatorname { max } \{ \alpha , \beta \}$. Therefore, only if $\gamma \leq - 1 / 2$ will $\Lambda _ { m } ^ { \alpha , \beta }$ have the optimal behaviour $\mathcal{O} (\operatorname { log } m )$. Denoting by $\Lambda _ { m } ^ { \alpha , \beta , r , s }$ the Lebesgue constant of the extended interpolation process, one has
  
<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/e/e120/e120270/e12027020.png" /></td> </tr></table>
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\begin{equation*} \Lambda _ { m } ^ { \alpha , \beta , r , s } \sim \operatorname { log }m \end{equation*}
  
 
if
 
if
  
<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/e/e120/e120270/e12027021.png" /></td> </tr></table>
+
\begin{equation*} \frac { \alpha } { 2 } + \frac { 1 } { 4 } \leq r < \frac { \alpha } { 2 } + \frac { 5 } { 4 }, \end{equation*}
  
<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/e/e120/e120270/e12027022.png" /></td> </tr></table>
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\begin{equation*} \frac { \beta } { 2 } + \frac { 1 } { 4 } \leq s < \frac { \beta } { 2 } + \frac { 5 } { 4 }. \end{equation*}
  
 
This technique has been extended to more general contexts and has led to the construction of many classes of optimal interpolation processes (see [[#References|[a1]]], [[#References|[a3]]] and the literature cited therein). These results have given important contributions to numerical [[Quadrature|quadrature]] and to collocation methods in the numerical solution of functional equations.
 
This technique has been extended to more general contexts and has led to the construction of many classes of optimal interpolation processes (see [[#References|[a1]]], [[#References|[a3]]] and the literature cited therein). These results have given important contributions to numerical [[Quadrature|quadrature]] and to collocation methods in the numerical solution of functional equations.
  
 
==Error estimation.==
 
==Error estimation.==
An efficient method for the practical estimation of the error of an interpolation process with respect to given nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027023.png" /> consists in imposing interpolation conditions at suitable additional nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027024.png" />. In particular, a natural choice are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027025.png" /> and nodes which interlace,
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An efficient method for the practical estimation of the error of an interpolation process with respect to given nodes $x _ { 1 } , \ldots , x _ { m }$ consists in imposing interpolation conditions at suitable additional nodes $y _ { 1 } , \ldots , y _ { n }$. In particular, a natural choice are $n = m + 1$ and nodes which interlace,
  
<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/e/e120/e120270/e12027026.png" /></td> </tr></table>
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\begin{equation*} a \leq y _ { 1 } < x _ { 1 } < y _ { 2 } < x _ { 2 } < \ldots < x _ { m } < y _ { m  + 1 } \leq b. \end{equation*}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027027.png" /> be the zeros of the [[Orthogonal polynomials|orthogonal polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027028.png" /> with respect to some weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027029.png" />. The zeros of the orthogonal polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027030.png" /> with respect to the same weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027031.png" /> have the interlacing property; orthogonal polynomials with respect to other weight functions have also been considered. For necessary and sufficient conditions for the convergence of the extended interpolation process, cf. [[#References|[a2]]] and the literature cited therein. The zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027032.png" /> of the associated [[Stieltjes polynomials|Stieltjes polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027033.png" /> lead to the Lagrange–Kronrod formulas, and they maximize the algebraic degree of the corresponding interpolatory quadrature formulas; the latter are the [[Gauss–Kronrod quadrature formula|Gauss–Kronrod quadrature formula]]. For the zeros of the Stieltjes polynomials, the interlacing property does not hold for general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120270/e12027034.png" />, but it is known for several important weight functions including the Legendre weight (see [[Stieltjes polynomials|Stieltjes polynomials]]). Error estimation by extended interpolation is an important tool for the numerical approximation of linear functionals.
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Let $x _ { 1 } , \ldots , x _ { m }$ be the zeros of the [[Orthogonal polynomials|orthogonal polynomials]] $P _ { m }$ with respect to some weight function $p$. The zeros of the orthogonal polynomials $P _ { m  + 1 }$ with respect to the same weight function $p$ have the interlacing property; orthogonal polynomials with respect to other weight functions have also been considered. For necessary and sufficient conditions for the convergence of the extended interpolation process, cf. [[#References|[a2]]] and the literature cited therein. The zeros $y _ { 1 } , \dots , y _ { m  + 1}$ of the associated [[Stieltjes polynomials|Stieltjes polynomials]] $E _ { m + 1} $ lead to the Lagrange–Kronrod formulas, and they maximize the algebraic degree of the corresponding interpolatory quadrature formulas; the latter are the [[Gauss–Kronrod quadrature formula|Gauss–Kronrod quadrature formula]]. For the zeros of the Stieltjes polynomials, the interlacing property does not hold for general $p$, but it is known for several important weight functions including the Legendre weight (see [[Stieltjes polynomials|Stieltjes polynomials]]). Error estimation by extended interpolation is an important tool for the numerical approximation of linear functionals.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Mastroianni,  "Uniform convergence of derivatives of Lagrange interpolation"  ''J. Comput. Appl. Math.'' , '''43''' :  2  (1992)  pp. 37–51</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Mastroianni,  "Approximation of functions by extended Lagrange interpolation"  R.V.M. Zahar (ed.) , ''Approximation and Computation'' , Birkhäuser  (1995)  pp. 409–420</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.O. Runck,  P. Vértesi,  "Some good point systems for derivatives of Lagrange interpolatory operators"  ''Acta Math. Hung.'' , '''56'''  (1990)  pp. 337–342</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Szabados,  "On the convergence of the derivatives of projection operators"  ''Analysis'' , '''7'''  (1987)  pp. 341–357</TD></TR></table>
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<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  G. Mastroianni,  "Uniform convergence of derivatives of Lagrange interpolation"  ''J. Comput. Appl. Math.'' , '''43''' :  2  (1992)  pp. 37–51</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  G. Mastroianni,  "Approximation of functions by extended Lagrange interpolation"  R.V.M. Zahar (ed.) , ''Approximation and Computation'' , Birkhäuser  (1995)  pp. 409–420</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  P.O. Runck,  P. Vértesi,  "Some good point systems for derivatives of Lagrange interpolatory operators"  ''Acta Math. Hung.'' , '''56'''  (1990)  pp. 337–342</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J. Szabados,  "On the convergence of the derivatives of projection operators"  ''Analysis'' , '''7'''  (1987)  pp. 341–357</td></tr>
 +
</table>

Latest revision as of 20:24, 5 December 2023

An interpolation process constructed from a given interpolation process by imposing additional interpolation conditions.

The method of additional nodes.

The method of additional nodes was introduced by J. Szabados in [a4] to approximate the derivatives of a function by means of the derivative of the Lagrange interpolating polynomial (cf. also Lagrange interpolation formula). For given interpolation nodes $x _ { 1 } < \ldots < x _ { m }$, the method consists in considering the interpolation process with respect to the nodes

where $y _ { i }$, $z_i$ are equidistant nodes between $a$ and $x_{1} $, respectively between $x _ { m }$ and $b$.

For $x _ { 1 } , \ldots , x _ { m }$ being the zeros of the Jacobi polynomial (cf. Jacobi polynomials) $P _ { m } ^{( \alpha , \beta )}$, $\alpha , \beta > - 1$, the Lebesgue constants $\Lambda _ { m } ^ { \alpha , \beta }$ of the Lagrange interpolating polynomials $p _ { m } ^ { \alpha , \beta }$ have the behaviour

\begin{equation*} \Lambda _ { m } ^ { \alpha , \beta } \sim \operatorname { max } \{ \operatorname { log } m , m ^ { \gamma + 1 / 2 } \}, \end{equation*}

where $\gamma = \operatorname { max } \{ \alpha , \beta \}$. Therefore, only if $\gamma \leq - 1 / 2$ will $\Lambda _ { m } ^ { \alpha , \beta }$ have the optimal behaviour $\mathcal{O} (\operatorname { log } m )$. Denoting by $\Lambda _ { m } ^ { \alpha , \beta , r , s }$ the Lebesgue constant of the extended interpolation process, one has

\begin{equation*} \Lambda _ { m } ^ { \alpha , \beta , r , s } \sim \operatorname { log }m \end{equation*}

if

\begin{equation*} \frac { \alpha } { 2 } + \frac { 1 } { 4 } \leq r < \frac { \alpha } { 2 } + \frac { 5 } { 4 }, \end{equation*}

\begin{equation*} \frac { \beta } { 2 } + \frac { 1 } { 4 } \leq s < \frac { \beta } { 2 } + \frac { 5 } { 4 }. \end{equation*}

This technique has been extended to more general contexts and has led to the construction of many classes of optimal interpolation processes (see [a1], [a3] and the literature cited therein). These results have given important contributions to numerical quadrature and to collocation methods in the numerical solution of functional equations.

Error estimation.

An efficient method for the practical estimation of the error of an interpolation process with respect to given nodes $x _ { 1 } , \ldots , x _ { m }$ consists in imposing interpolation conditions at suitable additional nodes $y _ { 1 } , \ldots , y _ { n }$. In particular, a natural choice are $n = m + 1$ and nodes which interlace,

\begin{equation*} a \leq y _ { 1 } < x _ { 1 } < y _ { 2 } < x _ { 2 } < \ldots < x _ { m } < y _ { m + 1 } \leq b. \end{equation*}

Let $x _ { 1 } , \ldots , x _ { m }$ be the zeros of the orthogonal polynomials $P _ { m }$ with respect to some weight function $p$. The zeros of the orthogonal polynomials $P _ { m + 1 }$ with respect to the same weight function $p$ have the interlacing property; orthogonal polynomials with respect to other weight functions have also been considered. For necessary and sufficient conditions for the convergence of the extended interpolation process, cf. [a2] and the literature cited therein. The zeros $y _ { 1 } , \dots , y _ { m + 1}$ of the associated Stieltjes polynomials $E _ { m + 1} $ lead to the Lagrange–Kronrod formulas, and they maximize the algebraic degree of the corresponding interpolatory quadrature formulas; the latter are the Gauss–Kronrod quadrature formula. For the zeros of the Stieltjes polynomials, the interlacing property does not hold for general $p$, but it is known for several important weight functions including the Legendre weight (see Stieltjes polynomials). Error estimation by extended interpolation is an important tool for the numerical approximation of linear functionals.

References

[a1] G. Mastroianni, "Uniform convergence of derivatives of Lagrange interpolation" J. Comput. Appl. Math. , 43 : 2 (1992) pp. 37–51
[a2] G. Mastroianni, "Approximation of functions by extended Lagrange interpolation" R.V.M. Zahar (ed.) , Approximation and Computation , Birkhäuser (1995) pp. 409–420
[a3] P.O. Runck, P. Vértesi, "Some good point systems for derivatives of Lagrange interpolatory operators" Acta Math. Hung. , 56 (1990) pp. 337–342
[a4] J. Szabados, "On the convergence of the derivatives of projection operators" Analysis , 7 (1987) pp. 341–357
How to Cite This Entry:
Extended interpolation process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extended_interpolation_process&oldid=13703
This article was adapted from an original article by Sven EhrichG. Mastroianni (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article