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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 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 $x _ { 1 } &lt; \ldots &lt; x _ { m }$, the method consists in considering the interpolation process with respect to the nodes

<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 $y _ { i }$, $z_i$ 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|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

\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 &lt; \frac { \alpha } { 2 } + \frac { 5 } { 4 }, \end{equation*}

\begin{equation*} \frac { \beta } { 2 } + \frac { 1 } { 4 } \leq s &lt; \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.

==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 } &lt; x _ { 1 } &lt; y _ { 2 } &lt; x _ { 2 } &lt; \ldots &lt; x _ { m } &lt; y _ { m  + 1 } \leq b. \end{equation*}

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====
<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>