Kergin interpolation

A form of interpolation providing a canonical polynomial of total degree $\leq m$ which interpolates a sufficiently differentiable function at $m + 1$ points in ${\bf R} ^ { n }$. (For $n > 1$ and $m > 1$ there is no unique interpolating polynomial of degree $\leq m$.)

More specifically, given $m + 1$ not necessarily distinct points in ${\bf R} ^ { n }$, $p = \{ p _ { 0 } , \dots , p _ { m } \}$, and $f$ an $m$-times continuously differentiable function on the convex hull of $p$, the Kergin interpolating polynomial $K _ { p } ( f )$ is of degree $\leq m$ and satisfies:

1) $K _ { p } ( f ) ( p _ { i } ) = f ( p _ { i } )$ for $i = 0 , \dots , m$; if a point $p_j$ is repeated $s \geq 2$ times, then $K _ { p } ( f )$ and $f$ have the same Taylor series up to order $s - 1$ at $p_j$;

2) for any constant-coefficient partial differential operator (cf. also Differential equation, partial) $Q ( \partial / \partial x )$ of degree $k \leq m$, one has $Q ( \partial / \partial x ) ( K _ { p } ( f ) - ( f ) )$ is zero at some point of the convex hull of any $k + 1$ of the points $\{ p _ { 0 } , \dots , p _ { m } \}$; furthermore, if $f$ satisfies an equation of the form $Q ( \partial / \partial x ) ( f ) \equiv 0$, then $Q ( \partial / \partial x ) ( K _ { p } ( f ) ) \equiv 0$;

3) for any affine mapping $\lambda : \mathbf R ^ { n } \rightarrow \mathbf R ^ { q }$ (cf. also Affine morphism) and $g$ an $m$-times continuously differentiable function on ${\bf R} ^ { q }$ one has $K _ { p } ( g \circ \lambda ) = K _ { \lambda ( p ) } ( g ) \circ \lambda$, where $\lambda ( p ) = \{ \lambda ( p _ { 0 } ) , \ldots , \lambda ( p _ { m } ) \}$;

4) the mapping $f \rightarrow K _ { p } ( f )$ is linear and continuous.

(In fact, 3)–4) already characterize the Kergin interpolating polynomial.)

The existence of $K _ { p }$ was established by P. Kergin in 1980 [a2]. For $n = 1$, $K _ { p }$ reduces to Lagrange–Hermite interpolation (cf. also Hermite interpolation formula; Lagrange interpolation formula).

An explicit formula for $K _ { p } ( f )$ was given by P. Milman and C. Micchelli [a3]. The formula shows that the coefficients of $K _ { p } ( f )$ are given by integrating derivatives of $f$ over faces in the convex hull of $p$. More specifically, let $S_r$ denote the simplex

\begin{equation*} S _ { r } = \left\{ ( v _ { 0 } , \dots , v _ { r } ) \in \mathbf{R} ^ { r + 1 } : v _ { j } \geq 0 , \sum _ { j = 0 } ^ { r } v _ { j } = 1 \right\} \end{equation*}

and use the notation

\begin{equation*} \int _ { [ p _ { 0 } \ldots p _ { r } ] } g = \int _ { S _ { r } } g ( v _ { 0 } p _ { 0 } + \ldots + v _ { r } p _ { r } ) d v _ { 1 } \ldots d v _ { r }. \end{equation*}

Then

\begin{equation*} K _ { p } (\, f ) = \sum _ { r = 0 } ^ { m } \int _ { [ p _ { 0 } \ldots p _ { r } ] } D _ { x - p _ { 0 }} \cdots D _ { x - p _ { r - 1 } }\,f, \end{equation*}

where $D _ { y } ( f )$ denotes the directional derivative of $f$ in the direction $y \in \mathbf{R} ^ { x }$.

Kergin interpolation also carries over to the complex case (as does Lagrange–Hermite interpolation), as follows. Let $\Omega \subset {\bf C} ^ { n }$ be a $\mathbf{C}$-convex domain (i.e. every intersection of $\Omega$ with a complex affine line is connected and simply connected, cf. also $\mathbf{C}$-convexity) and let $p = \{ p _ { 0 } , \dots , p _ { m } \}$ be $m + 1$ points in $\Omega$. For $f$ holomorphic on $\Omega$ there is a canonical analytic interpolating polynomial, $\kappa _ { p } ( f )$, of total degree $\leq m$ that satisfies properties corresponding to 1), 3), 4) above. If $\Omega$ is convex (identifying $\mathbf{C} ^ { n }$ with $\mathbf{R} ^ { 2 n }$), then $\kappa _ { p } ( f ) = K _ { p } ( \operatorname { Re } ( f ) ) + i K _ { p } ( \operatorname { Im } ( f ) )$. For general $\mathbf{C}$-convex domains (i.e. not necessarily real-convex), the formula for $\kappa _ { p } ( f )$, due to M. Andersson and M. Passare [a1], is analogous to the Milman–Micchelli formula above, but uses integration over singular chains.

There is a generalization of the Hermite remainder formula for Kergin interpolation if $\Omega$ is a bounded $\mathbf{C}$-convex domain with $C ^ { 2 }$ defining function $\rho$ and $f$ holomorphic in $\Omega$ and continuous up to the boundary $\partial \Omega$ of $\Omega$ [a1]. It is:

\begin{equation*} ( f - \kappa _ { p } ( f ) ) ( z ) = \end{equation*}

\begin{equation*} = \frac { 1 } { ( 2 \pi i ) ^ { n } } \int _ { \partial \Omega } \sum _ { | \alpha | + \beta = n - 1 } \left( \prod _ { j = 0 } ^ { m } \frac { \langle \rho ^ { \prime } ( \xi ) ,\, z - p _ { j } \rangle } { \langle \rho ^ { \prime } ( \xi ) ,\, \xi - p _ { j } \rangle } \right) \times \times \frac { f ( \xi ) \partial \rho ( \xi ) \wedge ( \overline { \partial } \partial \rho ( \xi ) ) ^ { n - 1 } } { \langle \rho ^ { \prime } ( \xi ) ,\, \xi - p \rangle ^ { \alpha } \langle \rho ^ { \prime } ( \xi ) ,\, \xi - z \rangle ^ { \beta + 1 } }, \end{equation*}

where $\alpha = ( \alpha_ 0 , \dots , \alpha _ { m } )$ is an $( m + 1 )$ multi-index, $\beta \geq 0$ is an integer, $\rho ^ { \prime } ( \xi ) = ( \partial \rho / \partial \xi _ { 1 } , \dots , \partial \rho / \partial \xi _ { n } )$ for $w \in \mathbf{C} ^ { n }$, $\langle z , w \rangle = \sum _ { j = 1 } ^ { x } z _ { j } w _ { j }$, and $\langle \rho ^ { \prime } ( \xi ) , \xi - p \rangle ^ { \alpha } = \prod _ { j = 0 } ^ { m } \langle \rho ^ { \prime } ( \xi ) , \xi - p _ { j } \rangle ^ { \alpha_j } $.

References

[a1]	M. Andersson, M. Passare, "Complex Kergin Interpolation" J. Approx. Th. , 64 (1991) pp. 214–225
[a2]	P. Kergin, "A natural interpolation of $C ^ { K }$ functions" J. Approx. Th. , 29 (1980) pp. 278–293
[a3]	C.A. Micchelli, P. Milman, "A formula for Kergin interpolation in $\mathbf{R} ^ { k }$" J. Approx. Th. , 29 (1980) pp. 294–296