Kergin interpolation
A form of interpolation providing a canonical polynomial of total degree which interpolates a sufficiently differentiable function at
points in
. (For
and
there is no unique interpolating polynomial of degree
.)
More specifically, given not necessarily distinct points in
,
, and
an
-times continuously differentiable function on the convex hull of
, the Kergin interpolating polynomial
is of degree
and satisfies:
1) for
; if a point
is repeated
times, then
and
have the same Taylor series up to order
at
;
2) for any constant-coefficient partial differential operator (cf. also Differential equation, partial) of degree
, one has
is zero at some point of the convex hull of any
of the points
; furthermore, if
satisfies an equation of the form
, then
;
3) for any affine mapping (cf. also Affine morphism) and
an
-times continuously differentiable function on
one has
, where
;
4) the mapping is linear and continuous.
(In fact, 3)–4) already characterize the Kergin interpolating polynomial.)
The existence of was established by P. Kergin in 1980 [a2]. For
,
reduces to Lagrange–Hermite interpolation (cf. also Hermite interpolation formula; Lagrange interpolation formula).
An explicit formula for was given by P. Milman and C. Micchelli [a3]. The formula shows that the coefficients of
are given by integrating derivatives of
over faces in the convex hull of
. More specifically, let
denote the simplex
![]() |
and use the notation
![]() |
Then
![]() |
where denotes the directional derivative of
in the direction
.
Kergin interpolation also carries over to the complex case (as does Lagrange–Hermite interpolation), as follows. Let be a
-convex domain (i.e. every intersection of
with a complex affine line is connected and simply connected, cf. also
-convexity) and let
be
points in
. For
holomorphic on
there is a canonical analytic interpolating polynomial,
, of total degree
that satisfies properties corresponding to 1), 3), 4) above. If
is convex (identifying
with
), then
. For general
-convex domains (i.e. not necessarily real-convex), the formula for
, 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 is a bounded
-convex domain with
defining function
and
holomorphic in
and continuous up to the boundary
of
[a1]. It is:
![]() |
![]() |
where is an
multi-index,
is an integer,
for
,
, and
.
References
[a1] | M. Andersson, M. Passare, "Complex Kergin Interpolation" J. Approx. Th. , 64 (1991) pp. 214–225 |
[a2] | P. Kergin, "A natural interpolation of ![]() |
[a3] | C.A. Micchelli, P. Milman, "A formula for Kergin interpolation in ![]() |
Kergin interpolation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kergin_interpolation&oldid=16059