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 functions" J. Approx. Th. , 29 (1980) pp. 278–293
[a3]	C.A. Micchelli, P. Milman, "A formula for Kergin interpolation in " J. Approx. Th. , 29 (1980) pp. 294–296