On the Structure of Kergin Interpolation for Points in General Position

L. Bos and S. Waldron

Abstract

For n + 1 points in IR^{d}, in general position, the Kergin polynomial interpolant of C^{n} functions may be extended to an interpolant of C^{d−1} functions. This results in an explicit set of reduced Kergin functionals naturally stratified by their dependence on certain directional derivatives of order k, 0 ≤ k ≤ d − 1. We show that the polynomials dual to the functionals depending on derivatives of order k are multi-ridge functions of d − k variables and moreover, that the polynomials dual to the purely interpolating functionals (k = 0) are always harmonic.

1 Introduction

Kergin interpolation is a natural multivariate extension of the Newton form of classical Hermite polynomial interpolation, introduced by Kergin in [4], and since then much studied (see, e.g., [6], [7], [1], [2]). It is most easily defined as follows.

Suppose that Θ is a sequence of n + 1 (possibly coincident) points in IR^{d}

Θ := [θ_{0}, θ_{1}, …, θ_{n}].

Then the Kergin interpolant of f ∈ C^{n}(IR^{d}) is the polynomial K_{Θ}f of degree n, defined by

K_{Θ}f(x) := ∑_{k=0}^{n} ∫_{[θ_{0},...,θ_{k}]} d^{k}f(x − θ_{0}, x − θ_{1}, …, x − θ_{k−1}).  (1.1)

Here d^{k}f denotes the k–th total derivative of f, and the linear functional

f ↦ ∫_{[θ_{0},...,θ_{k}]} f := \frac{1}{k! vol_{k}(S)} ∫_{S} f ∘ A,  (1.2)

where S is any k–simplex in IR^{s} with (k–dimensional) volume vol_{k}(S), and A : IR^{s} → IR^{d} is any affine map taking the k + 1 vertices of S onto the points θ_{0}, …, θ_{k}. The change of variables formula shows that (1.2) does not depend on the particular choice of S and A.

In the univariate case (1.1) reduces to the Newton form of the Hermite interpolant of f at Θ, via the Hermite–Genocchi formula for the divided difference of f at θ_{0}, …, θ_{k}, i.e.,

[θ_{0}, …, θ_{k}]f = ∫_{[θ_{0},...,θ_{k}]} D^{k}f.

Often it is convenient to write K_{Θ}f in terms of directional derivatives

K_{Θ}f(x) = ∑_{k=0}^{n} ∫_{[θ_{0},...,θ_{k}]} D_{x−θ_{0}}D_{x−θ_{1}} ⋯ D_{x−θ_{k−1}}f,

and to make a particular choice of S and A in (1.2), such as

∫_{[θ_{0},...,θ_{k}]} f = ∫_{0}^{1}∫_{0}^{s_{1}}⋯∫_{0}^{s_{k−1}}f(θ_{0}+s_{1}(θ_{1}−θ_{0})+⋯+s_{k}(θ_{k}−θ_{k−1})) ds_{k} ⋯ ds_{2} ds_{1}.

Amongst others, Kergin interpolation enjoys the following desirable properties:

\underline{Interpolation at Θ}: K_{Θ}f(θ_{j}) = f(θ_{j}), 0 ≤ j ≤ n.

\underline{Symmetry in Θ}: K_{Θ} does not depend on the ordering of the points in Θ.

\underline{Error Formula}: For each f ∈ C^{n+1}(IR^{d}),

f(x) − K_{Θ}f(x) = ∫_{[θ_{0},...,θ_{n}]} D_{x−θ_{0}}D_{x−θ_{1}} ⋯ D_{x−θ_{n}}f.

In particular, K_{Θ} is a linear projector onto Π_{n}(IR^{d}), the polynomials on IR^{d} of degree ≤ n.

\underline{Continuity and Coalescence}: For each f ∈ C^{n}(IR^{d}), the map

IR^{d×(n+1)} → Π_{n}(IR^{d}) : Θ ↦ K_{Θ}f

is continuous, and if θ_{0} = ⋯ = θ_{n} = a, then K_{Θ}f is the degree n Taylor polynomial of f centred at a.

\underline{Rolle’s Property}: For any homogeneous constant coefficient partial differential operator q(D) of degree k ≤ n,

∫_{[θ_{0},...,θ_{k}]} q(D)(f − K_{Θ}f) = 0.  (1.3)

\underline{PDE Faithfulness}: For any homogeneous constant coefficient partial differential operator q(D) of degree k ≤ n,

q(D)f = 0 ⇒ q(D)(K_{Θ}f) = 0.

The last property involves functions f : IR^{d} → IR which are constant on the translates of some affine subspace V of IR^{d} of codimension m. Since the collection of translates (also called cosets) IR^{d}/V := {x + V : x ∈ IR^{d}} partitions IR^{d},

f is constant on the translates of V

⟺ for every affine map A : IR^{d} → IR^{s} with kernel a translate of V, f = g ∘A

⟺ for some affine map A : IR^{d} → IR^{s} with kernel a translate of V, f = g ∘ A

where in each case g : ran(A) → IR is (well) defined on the range of A by g(Ax) := f(x). We will refer to such an f as a m–ridge or multi–ridge function. If m = 1, and A is taken to be a nonzero linear functional λ : x ↦ ⟨λ^{∗}, x⟩, λ^{∗} ∈ IR^{d}, then we obtain f = g ∘ λ, which is commonly referred as a ridge function, or plane wave.

\underline{Ridge Friendliness}: If f = g∘A is a multi–ridge function, where A : IR^{d} → IR^{s} is an affine map, and g ∈ C^{n}(ran(A)), then

K_{Θ}f = (K_{AΘ}g) ∘ A.

Because of the possibility of coalescence to a single point (Taylor interpolation), the Kergin interpolant is defined in general only for C^{n} functions. However, in the univariate case there is no smoothness requirement for distinct interpolation points, i.e., one has Lagrange interpolation to functions from C(IR).

This phenomenon extends to higher dimensional Kergin interpolants for points in general position in IR^{d}, which can be defined for functions of smoothness f ∈ C^{d−1}(IR^{d}). This extension can be described by a reduced set of linear functionals (interpolation conditions) that depend only on certain directional derivatives up to order d − 1, and their corresponding dual polynomials from Π_{n}(IR^{d}). A complete description of these reduced functionals is given in Theorem 2.3. It is natural to stratify them by the order of the derivative which they depend on, an “ascending” scale of complexity. In Theorem 2.4, we show the corresponding system of dual polynomials has a “descending” scale of complexity; precisely, that the dual polynomials to functionals involving derivatives of order k = 0, …, d − 1, are (d − k)–ridge functions, i.e., functions of only d − k variables. Finally, in Theorem 3.3 we show the surprising fact that the dual polynomials for the n + 1 point evaluation functionals are always harmonic.

The constant coefficient differential operators q(D) on IR^{d} used below are homogeneous of degree r ≥ 0, and are described in terms of the (homogeneous) polynomials q ∈ Π_{r}^{0}(IR^{d}) that correspond to them.

Lemma 1.1. Suppose that A : IR^{d} → IR^{s} is an affine map, with linear part L := A − A(0). If f is a multi–ridge function of the form f = g ∘ A, g ∈ C^{r}(ran(A)), then

∫_{Θ} q(D)f = ∫_{AΘ}(q ∘ L^{∗})(D)g,  ∀q ∈ Π_{r}^{0}(IR^{d}),  (1.4)

where L^{∗} : IR^{s} → IR^{d} is the adjoint of L.

Proof. We compute

D_{u}f(x) := lim_{t→0} \frac{g(A(x + tu)) − g(A(x))}{t}
= lim_{t→0} \frac{g(Ax + tLu) − g(A(x))}{t}
= (D_{Lu}g)(Ax),

and so (by iterating)

D_{u_{1}} ⋯ D_{u_{r}}f = (D_{Lu_{1}} ⋯ D_{Lu_{r}}g) ∘ A, ∀u_{1}, …, u_{r} ∈ IR^{d}.

Hence it follows from (1.2) that

∫_{Θ} D_{u_{1}} ⋯ D_{u_{r}}f = ∫_{Θ}(D_{Lu_{1}} ⋯ D_{Lu_{r}}g) ∘ A = ∫_{AΘ} D_{Lu_{1}} ⋯ D_{Lu_{r}}g. (1.5)

Since the differential operator q(D) = D_{u_{1}} ⋯ D_{u_{r}} corresponds to the homogeneous polynomial q(x) := (u_{1}^{T} x) ⋯ (u_{r}^{T} x), and D_{Lu_{1}} ⋯ D_{Lu_{r}} to

((Lu_{1})^{T} x) ⋯ ((Lu_{r})^{T} x) = (u_{1}^{T} L^{∗}x) ⋯ (u_{r}^{T} L^{∗}x) = q(L^{∗}x) = (q ∘ L^{∗})(x),

(1.5) can be expressed in the form (1.4).

2 The Ridge Structure of the Kergin Dual Polynomials

Definition 2.1. Suppose that V is a linear subspace of IR^{d}, with orthogonal complement V^{⊥}. Let Π_{k}(V) denote the subspace of Π_{k}(IR^{d}) consisting of multi– ridge polynomials which vary in the V directions, i.e., which are constant on translates of V^{⊥}, and Π_{k}^{0}(V) those which are in addition homogeneous of degree k.

If V has dimension m, then the natural identification with polynomials on IR^{m} gives

dim(Π_{k}(V)) = \binom{k + m}{m}, dim(Π_{k}^{0}(V)) = \binom{k + m − 1}{m − 1}.

It follows from the Rolle’s property and the error formula that K_{Θ} is the unique linear projector of C^{n}(IR^{d}) onto Π_{n}(IR^{d}) for which λ(K_{Θ}f) = λ(f), ∀λ ∈ Λ, where

Λ := ∑_{j=1}^{n+1} ∑_{{Ψ⊂Θ \atop #Ψ=j}} Span\{f ↦ ∫_{Ψ} q(D)f : q ∈ Π_{j−1}^{0}(IR^{d})\}.

Here Ψ ⊂ Θ, #Ψ = j indicates that Ψ is a subsequence of Θ of cardinality j. Henceforth we will refer to Λ as the (space of) interpolation conditions of K_{Θ}.

Definition 2.2. For Ψ a sequence of k + 1 affinely independent points in IR^{d}, let Λ_{Ψ}^{i} be the space of (continuous) linear functionals C^{i}(conv(Ψ)) → IR

Λ_{Ψ}^{i} := {f ↦ ∫_{Ψ} q(D)f : q ∈ Π_{i}^{0}(aff(Ψ)^{⊥})},  i = 0, 1, ….  (2.1)

The q(D) occuring above consist of all i–th order derivatives normal to aff(Ψ), i.e., are spanned by D_{n_{1}}D_{n_{2}} ⋯ D_{n_{i}}, where the n_{1}, n_{2}, …, n_{i} are directions orthogonal to aff(Ψ).

The following decomposition of the interpolation conditions Λ was given in [8].

Theorem 2.3. (Reduced functionals) Suppose the points in Θ are in general position. Then the space of interpolation conditions for K_{Θ} can be written as a direct sum

Λ = ⨁_{j=1}^{d} Λ_{j−1},  Λ_{j−1} := ⨁_{{Ψ⊂Θ \atop #Ψ=j}} Λ_{Ψ},  (2.2)

where the linear functionals

Λ_{Ψ} := Λ_{Ψ}^{j−1} = {f ↦ ∫_{Ψ} q(D)f : q ∈ Π_{j−1}^{0}(aff(Ψ)^{⊥})}

are continuous on C^{d−1}(IR^{d}).

This was proved by using the identity

∫_{[Θ,v,w]} D_{v−w}f = ∫_{[Θ,v]} f − ∫_{[Θ,w]} f,

to reduce the order of the derivatives occuring in the interpolation conditions as much as possible, then doing a dimension count of the conditions so obtained. This result implies K_{Θ} has a continuous extension to C^{d−1}(IR^{d}) → Π_{n}(IR^{d}).

Theorem 2.4. (Dual polynomials) Let P_{Ψ}^{Θ} be the subspaces of Π_{n}(IR^{d}) dual to those of (2.2), i.e., satisfying dim(Λ_{Ψ}|_{P_{Ψ}^{Θ}}) = dim(P_{Ψ}^{Θ}), and Λ_{Ψ_{1}}(P_{Ψ_{2}}^{Θ}) = 0, Ψ_{1} ≠ Ψ_{2}, which implies

Π_{n}(IR^{d}) = ⨁_{j=1}^{d} ⨁_{{Ψ⊂Θ \atop #Ψ=j}} P_{Ψ}^{Θ}.  (2.3)

Then P_{Ψ}^{Θ}, #Ψ = j, consists of (d − j + 1)–ridge functions in the directions aff(Ψ)^{⊥}, i.e.,

P_{Ψ}^{Θ} ⊂ Π_{n}(aff(Ψ)^{⊥}).

Proof. Fix Ψ_{0} ⊂ Θ, #Ψ_{0} = k + 1, and consider the space of functionals Λ_{Ψ_{0}}. Let A : IR^{d} → IR^{d} be the orthogonal projection onto V := aff(Ψ_{0})^{⊥}. Then, by the ridge–friendliness of Kergin interpolation,

K_{Θ}(g ∘ A) = (K_{AΘ}g) ∘ A,  ∀g ∈ C^{d−1}(V).  (2.4)

Here K_{AΘ}g is the Kergin interpolant for the sequence of “projected” points AΘ ⊂ V, which is defined for g ∈ C^{d−1}(V) via (1.1). The k + 1 points of Ψ_{0} project onto a single point, say ψ_{0}, and the remaining projected points are distinct (since Θ is in general position).

By Lemma 1.1, and A = A^{∗} (orthogonal projections are self adjoint), the linear functionals in Λ_{Ψ} restricted to multi–ridge functions f = g ∘ A have the form

f ↦ ∫_{Ψ} q(D)f = ∫_{AΨ}(q ∘ A)(D)g,  q ∈ Π_{j−1}^{0}(aff(Ψ)^{⊥}).  (2.5)

Let Λ_{Ψ}^{|} denote the corresponding space of projected interpolation conditions

C^{d−1}(V) → IR : g ↦ ∫_{AΨ}(q ∘ A)(D)g, q ∈ Π_{j−1}^{0}(aff(Ψ)^{⊥}).

Then, by (2.4), the interpolation conditions of K_{AΘ} are Λ^{|} := ∑_{Ψ} Λ_{Ψ}^{|}, and we now show

Λ^{|} = Λ_{Ψ_{0}}^{|} ⨁ ∑_{{Ψ⊂Θ \atop Ψ≠Ψ_{0}}} Λ_{Ψ}^{|},  dim(Λ_{Ψ_{0}}^{|}|_{Π_{n}(IR^{d})|_{V}}) = dim(Λ_{Ψ_{0}}).  (2.6)

Firstly, by (2.5), the space Λ_{Ψ_{0}}^{|} consists of the linear functionals

g ↦ \frac{1}{k!}q(D)g(ψ_{0}), q ∈ Π_{k}^{0}(V),

and so has

dim(Λ_{Ψ_{0}}^{|}|_{Π_{n}(IR^{d})|_{V}}) = dim(Π_{k}^{0}(V)) = dim(Λ_{Ψ_{0}}).

Let L be any orthogonal projection onto a 1–dimensional subspace of V which maps only the points Ψ_{0} onto Lψ_{0} = L(Aψ_{0}). Since Θ is in general position all but a finite number of the possible L will have this property. By (1.4) and the Hermite–Genocchi formula, a linear functional from Λ_{Ψ}^{|} applied to a ridge function of the form f = g ∘ L is a multiple of the divided difference of the univariate function g ∈ C^{d−1}(ran(L)) at the points LΨ. Hence, if

λ^{|} ∈ Λ_{Ψ_{0}}^{|} ⋂ ∑_{{Ψ⊂Θ \atop Ψ≠Ψ_{0}}} Λ_{Ψ}^{|},

then λ^{|}(f) = 0, since the only LΨ, Ψ ≠ Ψ_{0} that involve the point Lψ_{0} k + 1 times are those with Ψ ⊃ Ψ_{0}. By the fundamentality of such ridge functions f = g ∘ L, we have λ^{|} = 0.

Let Q be the subspace of Π_{n}(IR^{d})|_{V} dual to the first summand of (2.6), i.e., satisfying

dim(Λ_{Ψ_{0}}^{|}|_{Q}) = dim(Λ_{Ψ_{0}}), Λ_{Ψ}^{|}(Q) = 0, ∀Ψ ≠ Ψ_{0}.

Then P_{Ψ_{0}}^{Θ} = Q ∘ A ⊂ Π_{n}(aff(Ψ_{0})^{⊥}), since

dim(Λ_{Ψ_{0}}|_{Q∘A}) = dim(Λ_{Ψ_{0}}^{|}|_{Q}) = dim(Λ_{Ψ_{0}}) ∀Ψ ≠ Ψ_{0}

and

Λ_{Ψ}(Q ∘ A) = Λ_{Ψ}^{|}(Q) = 0, ∀Ψ ≠ Ψ_{0}.

Remark. Kergin interpolation can also be defined for points in CI ^{d}, via an extension of the functional (1.2), cf. [1]. This inherits all the properties of real Kergin interpolation, in a natural way, and a simple check shows the dual polynomials for Kergin interpolation to points in general position in CI ^{d} have the multi–ridge structure of Theorem 2.4.

Example. Consider the interpolation conditions Λ_{Ψ_{0}}, where #Ψ_{0} = d. This space is 1–dimensional, with a basis given by

λ : f ↦ ∫_{Ψ_{0}} D_{n}^{d−1}f,

where n is a unit normal to the hyperplane aff(Ψ_{0}). Let p = p_{λ} ∈ P_{Ψ_{0}}^{Θ} be the polynomial dual to λ, then by our proof, this is the ridge function

p(x) = g(⟨n, x⟩), g := (· − ψ_{0})^{d−1} ∏_{θ∈Θ\Ψ_{0}}(· − ⟨n, θ⟩), ψ_{0} := ⟨n, Ψ_{0}⟩.

where g is the Hermite interpolant to the data

g(⟨n, θ⟩) = 0, θ ∈ Θ \ Ψ_{0}, (n − d conditions)

and

g(ψ_{0}) = Dg(ψ_{0}) = ⋯ = D^{d−2}g(ψ_{0}) = 0, \frac{D^{d−1}g(ψ_{0})}{(d − 1)!} = 1 (d conditions).

3 Harmonicity of the dual polynomials for point evaluation

Let Δ denote the Laplace operator in IR^{d}. It is well known that the map

Δ|_{Π_{n}(IR^{d})} : Π_{n}(IR^{d}) → Π_{n−2}(IR^{d}) : f ↦ Δf

is onto, and so its kernel IH_{n}(IR^{d}), the harmonic polynomials of degree ≤ n in IR^{d}, has

dim(IH_{n}(IR^{d})) = \binom{n + d}{d} − \binom{n − 2 + d}{d}.

Since dim(IH_{n}(IR^{d})) < dim(Π_{n}(IR^{d})) = dim Λ, n ≥ 2, the Kergin interpolation conditions Λ of (2.2) are not linearly independent over IH_{n}(IR^{d}), i.e., as functionals restricted to IH_{n}(IR^{d}). We now describe these dependencies, which also hold for all harmonic functions. These then imply that the dual polynomials to point evaluations are harmonic for d > 1.

For a sequence of points Ψ in IR^{d}, denote by Δ_{Ψ}^{⊥} the Laplacian in the directions orthogonal to aff(Ψ), i.e.,

Δ_{Ψ}^{⊥} = q_{Ψ}^{⊥}(D) := D_{n_{1}}^{2} + D_{n_{2}}^{2} + ⋯ + D_{n_{d−k}}^{2}, q_{Ψ}^{⊥} ∈ Π_{2}^{0}(aff(Ψ)^{⊥}),

where n_{1}, …, n_{d−k} is any orthonormal basis for aff(Ψ)^{⊥}. Denote by conv(Ψ) the convex hull of the points in Ψ.

Lemma 3.1. Suppose that Ψ is a sequence of k + 1 affinely independent points in IR^{d}, and let λ ∈ Λ_{Ψ}^{i}, i ≥ 2, be the linear functional given by

λ(f) := ∫_{Ψ} Δ_{Ψ}^{⊥}p(D)f,  p ∈ Π_{i−2}^{0}(aff(Ψ)^{⊥}).  (3.1)

Then

λ|_{IH} ∈ ∑_{{Ψ̃⊂Ψ \atop #Ψ̃=k}} Λi−1 ; Ψ̃|_{IH},  IH := \{f ∈ C^{i}(conv(Ψ)) : Δf = 0\}.  (3.2)

Proof. Let Δ_{Ψ} denote the Laplacian in directions parallel to aff(Ψ), i.e., split Δ = Δ_{Ψ} + Δ_{Ψ}^{⊥}. Suppose that f is harmonic. Then

Δp(D)f = p(D)Δf = 0 ⇒ Δ_{Ψ}^{⊥}p(D)f = −Δ_{Ψ}p(D)f,

and so, by (1.2),

λ(f) = ∫_{Ψ} Δ_{Ψ}^{⊥}p(D)f = − ∫_{Ψ} Δ_{Ψ}p(D)f = −\frac{1}{k! vol_{k}(conv(Ψ))} ∫_{conv(Ψ)} Δ_{Ψ}p(D)f.

Now, by Green’s theorem,

∫_{conv(Ψ)} Δ_{Ψ}g = ∫_{∂(conv(Ψ))} D_{n}g = ∑_{{Ψ̃⊂Ψ \atop #Ψ̃=k}} ∫_{conv(Ψ̃)} D_{n}g, ∀g ∈ C^{2}(conv(Ψ)),

where n is the outward normal to the boundary of the k–simplex conv(Ψ), which consists of the faces conv(Ψ̃). Hence, using (1.2), λ|_{IH} can be expressed as a linear combination of functionals

λ(f) = ∑_{{Ψ̃⊂Ψ \atop #Ψ̃=k}} cΨ̃ ∫Ψ̃ D_{n}p(D)f, cΨ̃ := −\frac{vol_{k−1}(conv(Ψ̃))}{k vol_{k}(conv(Ψ))},

each of which belongs to Λi−1 ; Ψ̃, since p ∈ Π_{i−2}^{0}(aff(Ψ)^{⊥}) ⇒ p ∈ Π_{i−2}^{0}(aff(Ψ̃)^{⊥}).

There are also linear dependencies, of a different type, between the functionals Λ_{Ψ}^{1} over harmonic functions.

Lemma 3.2. Suppose that Ψ is a sequence of d + 1 affinely independent points in IR^{d}. Then, for f ∈ C^{1}(conv(Ψ)) harmonic,

∑_{{Ψ̃⊂Ψ \atop #Ψ̃=d}} aΨ̃ ∫Ψ̃ D_{n}f = 0, aΨ̃ := (d − 1)!vol_{d−1}(conv(Ψ̃)),

where n denotes the outward normal to conv(Ψ̃) as a face of the simplex conv(Ψ).

Proof. By Green’s Theorem,

∑_{{Ψ̃⊂Ψ \atop #Ψ̃=d}} aΨ̃ ∫Ψ̃ D_{n}f = ∑_{{Ψ̃⊂Ψ \atop #Ψ̃=d}} ∫_{conv(Ψ̃)} D_{n}f = ∫_{∂(conv(Ψ))} D_{n}f = ∫_{conv(Ψ)} Δf = 0.

In [2], using the identification of IR^{2} with CI , it was shown that the dual polynomials to point evaluation for Kergin interpolation to points in general position in IR^{2} are given by the real part of dual polynomials for univariate complex Lagrange interpolation to the points in CI , and so are harmonic. We now show this harmonicity extends to higher dimensions (including the odd dimensions!).

Theorem 3.3. (Harmonicity). For Kergin interpolation to Θ in general position in IR^{d}, d ≥ 2, the dual polynomials to point evaluation are harmonic, i.e., ΔP_{[ψ]}^{Θ} = 0, ∀ψ ∈ Θ.

Proof. We begin by giving an alternative proof of the bivariate result, which is easily generalised to cover the so called scale of mean value interpolants (see, e.g., [8]).

In the bivariate case, the reduced interpolation conditions Λ of (2.2) consist of n + 1 point evaluations Λ_{0}, and \binom{n+1}{2} first derivative functionals Λ_{1}. By Lemma 3.2, for each triple of points Ψ := [ψ_{1}, ψ_{2}, ψ_{3}] ⊂ Θ, the associated first derivative functionals have a linear dependency over harmonic functions given by

∥ψ_{2} − ψ_{1}∥ ∫_{[ψ_{1},ψ_{2}]} D_{n}f + ∥ψ_{3} − ψ_{2}∥ ∫_{[ψ_{2},ψ_{3}]} D_{n}f + ∥ψ_{1} − ψ_{3}∥ ∫_{[ψ_{3},ψ_{1}]} D_{n}f = 0.

Since each pair of points is in a triple containing a given point θ_{0}, this implies the n first derivative functionals that involve θ_{0} and another point span Λ_{1} over IH_{n}(IR^{2}). But,

dim(Λ|_{IH_{n}(IR^{2})}) = dim(IH_{n}(IR^{2})) = 2n + 1 = (n + 1) + n = dim(Λ_{0}) + n,

and so these n functionals must be a basis for Λ_{1} over IH_{n}(IR^{2}), which when appended by the (n + 1) point evaluations Λ_{0} forms a dual basis for IH_{n}(IR^{2}). Let h ∈ IH_{n}(IR^{2}) be the harmonic polynomial which is 1 at a point ψ of Θ, 0 at the others, and is annihilated by the n first derivative functionals above, and hence all of Λ_{1}. Then this h satisfies the conditions that characterise the Kergin dual polynomial to point evaluation at ψ, which is therefore harmonic.

Now the case d ≥ 3. The idea is the same, except we use the linear dependencies of Lemma 3.1. Let us count how many different dependencies this gives. Since Θ is in general position, for 2 ≤ k < d, each of the \binom{n+1}{k+1} subsequences Ψ of k + 1 points is affinely independent, and the subspace Λ_{Ψ}^{dep} of Λ_{Ψ} = Λ_{Ψ}^{k} of functionals of the form (3.1) has

dim(Λ_{Ψ}^{dep}) = dim(Π_{k−2}^{0}(aff(Ψ)^{⊥})) = \binom{k − 2 + (d − k − 1)}{d − k − 1} = \binom{d − 3}{d − k − 1}.

This gives a total number of dependencies

∑_{k=2}^{d−1} \binom{d − 3}{d − k − 1}\binom{n + 1}{k + 1} = \binom{n − 2 + d}{d},  (3.3)

as the above sum represents the number of ways of choosing d objects from n − 2 + d objects after having split them into two groups, one of size d − 3 and the other of size n + 1. Let Λ_{Ψ}^{∗} be a complement of Λ_{Ψ}^{dep} in Λ_{Ψ}, i.e., write Λ_{Ψ} = Λ_{Ψ}^{∗} ⊕ Λ_{Ψ}^{dep}. Then, by (3.2),

Λ_{Ψ}|_{IH} = Λ_{Ψ}^{∗}|_{IH}+Λ_{Ψ}^{dep}|_{IH} ⊂ Λ_{Ψ}^{∗}|_{IH}+Λ_{k−1}|_{IH}, IH := {f ∈ C^{d−1}(conv(Θ)) : Δf = 0},

and so the space of functionals

Λ^{∗} := Λ_{0} ⊕ Λ_{1} ⊕ ⨁_{j=3}^{d} ⨁_{{Ψ⊂Θ \atop #Ψ=j}} Λ_{Ψ}^{∗},

spans Λ over IH. By (3.3),

dim(Λ^{∗}) = \binom{n + d}{d} − \binom{n − 2 + d}{d} = dim(IH_{n}(IR^{d})),

and so Λ^{∗} is dual to IH_{n}(IR^{d}). Let h ∈ IH_{n}(IR^{d}) be the harmonic polynomial which is 1 at a point ψ of Θ, 0 at the others, and is annihilated by the other summands in the definition of Λ^{∗}, and hence by Λ_{1}, …, Λ_{d−1}. Then this h satisfies the conditions that characterise the Kergin dual polynomial to point evaluation at ψ, which is therefore harmonic.

Remark. The above result can be generalised to the mean value interpolation of [3]. Briefly, for points in general position (see [8]), there is a family of linear projectors

H_{Θ}^{(m)} : C^{n−m−1}(IR^{d}) → Π_{n−m−1}(IR^{d}), 0 ≤ m ≤ d − 1,

called the scale of mean value interpolations, determined by the interpolation conditions

Λ = ⨁_{j=m+1}^{d} Λ_{j−m−1}^{(m)} = ⨁_{j=m+1}^{d} ⨁_{{Ψ⊂Θ \atop #Ψ=j}} Λ_{Ψ}^{j−m−1},

which contain Kergin interpolation as the special case K_{Θ} = H_{Θ}^{(0)}. A modification of the above argument shows that for all mean value interpolation operators, except Hakopian interpolation, when m = d − 1, the dual polynomials to the functionals Λ_{0}^{(m)} (which involve the function, but no derivatives) are harmonic.

References

[1] M. Andersson, M. Passare: Complex Kergin interpolation, J. Approx. Theory, 64 (1991), 214–225.

[2] L. Bos, J.-P. Calvi: Kergin interpolants at the roots of unity approximate C^{2} functions, J. Anal. Math. 72 (1997), 203–221.

[3] T. N. T. Goodman: Interpolation in minimum semi-norm, and multivariate B-splines, J. Approx. Theory 37 (1983), 212–223.

[4] P. Kergin: A natural interpolation of C^{k} functions, J. Approx. Theory 29 (1980), 278–293.

[5] V. Ya. Lin, A. Pinkus: Fundamentality of ridge functions, J. Approx. Theory 75(3) (1993), 295–311.

[6] C. A. Micchelli: A constructive approach to Kergin interpolation in IR^{k} : multivariate B-splines and Lagrange interpolation, Rocky Mountain J. Math. 10 (1980), 485–497.

[7] C. A. Micchelli, P. Milman: A formula for Kergin interpolation in IR^{k}, J. Approx. Theory 29 (1980), 294–296.

[8] S. Waldron: Mean value interpolation for points in general position, preprint

Addresses:

L. Bos Department of Mathematics and Statistics University of Calgary Calgary, Alberta, Canada T2N 1N4

S. Waldron Department of Mathematics University of Auckland Private Bag 92019 Auckland, New Zealand