                              29 January 2002

The diagonalisation of the multivariate Bernstein operator

Shaun Cooper and Shayne Waldron

Mathematics, Massey University, Albany Campus, Private Bag 102904, Auckland, New Zealand email: S.Cooper@massey.ac.nz

Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand e–mail: waldron@math.auckland.ac.nz (http://www.math.auckland.ac.nz/˜ waldron)

ABSTRACT

Let B_{n} be the multivariate Bernstein operator of degree n for a simplex in IR^{s}. In this paper we show that B_{n} is diagonalisable with the same eigenvalues as the univariate Bernstein operator, i.e.,

λ_{k}^{(n)} := \frac{n!}{(n − k)!} \frac{1}{n^{k}} , k = 1, …, n, 1 = λ_{1}^{(n)} > λ_{2}^{(n)} > ⋯ > λ_{n}^{(n)} > 0,

and we describe the corresponding eigenfuctions and their properties.

Since B_{n} reproduces only the linear polynomials, these are the eigenspace for λ_{1}^{(n)} = 1. For k > 1, the λ_{k}^{(n)}–eigenspace consists of polynomials of exact degree k, which are uniquely determined by their leading term. These are described in terms of the substitution of the barycentric coordinates (for the underlying simplex) into elementary eigenfunctions. It turns out that there are eigenfunctions of every degree k which are common to each B_{n}, n ≥ k, for sufficiently large s. The limiting eigenfunctions and their connection with orthogonal polynomials of several variables is also considered.

Key Words: multivariate Bernstein operator, diagonalisation, eigenvalues, eigenfunctions, total positivity, Stirling numbers, Jacobi polynomials, semigroup, quasi–interpolant

AMS (MOS) Subject Classifications: primary 41A10, 15A18, 38B42, secondary 33C45, 41A36

1. Introduction

This paper is the multivariate counterpart of Cooper and Waldron [CW00], which gave the spectral decomposition of the univariate Bernstein operator and applications. Here we show that the multivariate Bernstein operator B_{n} for a simplex in IR^{s} is also diagonalisable, with the same eigenvalues, namely

λ_{k}^{(n)} := \frac{n!}{(n − k)!} \frac{1}{n^{k}} , k = 1, …, n,  1 = λ_{1}^{(n)} > λ_{2}^{(n)} > ⋯ > λ_{n}^{(n)} > 0,  (1.1)

and give an explicit formula for the eigenfunctions. The paper is set out as follows.

In the second half of this section, we define B_{n}, establish notation and give some technical results. The notation used is based on that of de Boor [B87], which indexes the barycentric coordinates by the vertices they correspond to, rather than imposing some ordering on them. This leads to a compact notation which simplifies the calculations and reveals the underlying geometry.

In Section 2, we give the diagonalisation and describe its symmetries. Since B_{n} reproduces only the linear polynomials, these give the eigenspace for λ_{1}^{(n)} = 1. For k > 1, the λ_{k}^{(n)}–eigenspace is no longer 1–dimensional, as it is in the univariate case. It consists of polynomials of exact degree k which are uniquely determined by their leading term, and for which an explicit formula is provided.

In Section 3, we show that B_{n}f takes a simplified form when f is certain ridge–type functions. This is used to describe the eigenfunctions of B_{n} in terms of the substitution of the barycentric coordinates (for the underlying simplex) into elementary eigenfunctions. It turns out that there are eigenfunctions of every degree k which are common to each B_{n}, n ≥ k, for sufficiently large s.

In Section 4, we show, as in the univariate case, that the eigenfunctions (with fixed leading term) converge as n → ∞. We describe these limiting eigenfunctions and their connection with orthogonal polynomials of several variables.

In Section 5, we give an interesting result about B_{n} applied to certain shifted factorials, and some consequences of it.

We conclude with some comments about those aspects of the univariate theory which have not been extended to the multivariate setting. The appendix contains a list of the elementary eigenfunctions.

We now give the definitions which will be used throughout.

Definitions

Let V be a set of s + 1 affinely independent points in IR^{s}, i.e., the vertices of an s–simplex which we denote by T. Denote by ξ = (ξ_{v})_{v∈V} the corresponding barycentric coordinates, i.e., the unique linear polynomials which satisfy

∑_{v∈V} ξ_{v}(x) = 1,  ∑_{v∈V} ξ_{v}(x)v = x, ∀x ∈ IR^{s}.

We will use standard multi–index notation with indices from ZZ_{+}^{V} and ZZ_{+}^{s}, so, for example,

ξ^{α} := ∏_{v∈V} ξ_{v}^{α(v)}, α ∈ ZZ_{+}^{V}, β! := β_{1}!β_{2}! ⋯ β_{s}!, β ∈ ZZ_{+}^{s}.

The Bernstein operator of degree n for the simplex T with vertices V is defined by

B_{n,V} f := ∑_{{|α|=n \atop α∈Z Z^{V}_{+}}} \binom{n}{α}ξ^{α}f(v_{α}),  ∀f ∈ C(T),  (1.2)

where

\binom{n}{α} := \frac{n!}{α!(n − |α|)!}, v_{α} := ∑_{v∈V} \frac{α(v)}{|α|}v ∈ T.

For V = {v_{0}, v_{1}, …, v_{s}}, this can be rewritten as

B_{n,V} f = ∑_{k=0}^{n} ∑_{{|α|=k \atop α∈Z Z^{s}_{+}}} \binom{n}{α}ξ_{v_{1}}^{α_{1}} ⋯ ξ_{v_{s}}^{α_{s}}ξ_{v_{0}}^{n−k} f(\frac{α_{1}v_{1} + ⋯ + α_{s}v_{s} + (n − k)v_{0}}{n} ).  (1.3)

If T is the standard simplex in IR^{s}, i.e.,

V = {0, e_{1}, …, e_{s}} (e_{i} the standard basis vectors for IR^{s})

then (1.3) becomes (cf [L53:(13),p. 51])

B_{n}f(x) = ∑_{k=0}^{n} ∑_{|α|=k} \binom{n}{α}x_{1}^{α_{1}} ⋯ x_{s}^{α_{s}}(1 − x_{1} − ⋯ − x_{s})^{n−k} f(\frac{α_{1}}{n} , …, \frac{α_{s}}{n} ).  (1.4)

It is well known that B_{n,V} maps onto Π_{n}(IR^{s}) the polynomials of degree n in IR^{s}.

Let S_{V} be the symmetry group of the simplex T with vertices V, i.e., the group of affine transformations which map T onto T. This is (isomorphic to) the symmetric group on the s + 1 vertices V since an affine map IR^{s} → IR^{s} is uniquely determined by its action on s + 1 affinely independent points. It follows from (1.2) that

B_{n,V} (f ∘ A) = (B_{n,V} f) ∘ A,  ∀f ∈ C(T), ∀A ∈ S_{V} .  (1.5)

We let S_{V} act on functions p ∈ C(T) and linear functionals μ defined on C(T) in the usual way, i.e., for A ∈ S_{V}

A · p := p ∘ A^{−1}, (A · μ)(f) := μ(f ∘ A), ∀f ∈ C(T).

For β ∈ ZZ_{+}^{V} with |β| = k and h > 0, define the multivariate shifted factorial by

[ξ]_{h}^{β} := ∏_{v∈V} [ξ_{v}]_{h}^{β(v)} ∈ Π_{k}, [ξ_{v}]_{h}^{β(v)} := ξ_{v}(ξ_{v} − h)(ξ_{v} − 2h) ⋯ (ξ_{v} − (β(v) − 1)h).

and the multivariate Stirling numbers of the first kind from the univariate ones by

S(β, α) := ∏_{v∈V} S(β(v), α(v)),  α ≤ β.  (1.6)

These are related by

ξ^{β} = ∑_{α≤β} S(β, α)h^{|β−α|}[ξ]_{h}^{α},  ∀β ∈ ZZ_{+}^{V}, ∀h > 0,  (1.7)

which follows from the univariate result

ξ_{v}^{β(v)} = ∑_{α(v)=0}^{β(v)} S(β(v), α(v))h^{β(v)−α(v)}[ξ_{v}]_{h}^{β(v)−α(v)},

by the calculation

ξ^{β} = ∏_{v∈V} ( ∑_{α(v)=0}^{β(v)} S(β(v), α(v))h^{β(v)−α(v)}[ξ_{v}]_{h}^{α(v)})

= ∑_{α≤β} ( ∏_{v∈V} S(β(v), α(v))h^{β(v)−α(v)}[ξ_{v}]_{h}^{α(v)})

= ∑_{α≤β} S(β, α)h^{|β−α|}[ξ]_{h}^{α}.

A special case of Theorem 5.2 of Section 5 is that

B_{n,V} ([ξ]_{1/n}^{β}) = λ_{k}^{(n)}ξ^{β}, |β| = k ≤ n,

where λ_{k}^{(n)} are the eigenvalues defined by (1.1).

For α ∈ ZZ_{+}^{V} with α(v_{0}) = 0 and h > 0, let Δ_{h,v_{0}}^{α} be the multivariate difference operator defined by

Δ_{h,v_{0}}^{α}f := ∑_{β≤α ; {β(v_{0})=0 \atop β∈Z Z^{V}_{+}}} \binom{α}{β}(−1)^{|α−β|}f(· + h ∑_{v∈V} β(v)(v − v_{0})),  \binom{α}{β} := \frac{α!}{β!(α − β)!}. (1.8)

Let e_{v} ∈ ZZ_{+}^{V} be the multi–index with

e_{v}(w) := \{ \begin{matrix}1, & w = v; \\ 0, & otherwise. \end{matrix}

Lemma 1.9. Let α ∈ ZZ_{+}^{V} with α(v_{0}) = 0 and h > 0. Then Δ_{h,v_{0}}^{α} satisfies

Δ_{h,v_{0}}^{e_{v}}f = f(· + h(v − v_{0})) − f,  Δ_{h,v_{0}}^{α+β}f = Δ_{h,v_{0}}^{α}(Δ_{h,v_{0}}^{β}f),  (1.10)

and

Δ_{h,v_{0}}^{α}[ξ]_{h}^{β} = \{ \begin{matrix}\frac{β!}{(β−α)!}h^{|α|}[ξ]_{h}^{β−α}, & α ≤ β; \\ 0, & otherwise. \end{matrix}  (1.11)

Proof: From the definition (1.8), we have

Δ_{h,v_{0}}^{e_{v}} = f(· + h(v − v_{0})) − f.

Since the first order differences Δ_{h,v_{0}}^{e_{v}} and Δ_{h,v_{0}}^{e_{w}} commute, to prove the second part of (1.10) it is sufficient to show

Δ_{h,v_{0}}^{α} = ∏_{{v∈V \atop v≠v_{0}}} \underbrace{Δ_{h,v_{0}}^{e_{v}} ⋯ Δ_{h,v_{0}}^{e_{v}}}_{α(v) times},  (1.12)

by induction on |α|. For |α| = 0 we have Δ_{h,v_{0}}^{α}f = f. Now suppose α(w) > 0, for some w ∈ V, then by the inductive hypothesis

∏_{{v∈V \atop v≠v_{0}}} \underbrace{Δ_{h,v_{0}}^{e_{v}} ⋯ Δ_{h,v_{0}}^{e_{v}}}_{α(v) times} f = Δ_{h,v_{0}}^{e_{w}}Δ_{h,v_{0}}^{α−e_{w}}f

= Δ_{h,v_{0}}^{e_{w}} ∑_{β≤α−ew ; {β(v_{0})=0 \atop β∈Z Z^{V}_{+}}} \binom{α − e_{w}}{β}(−1)^{|α−e_{w}−β|}f(· + h ∑_{v∈V} β(v)(v − v_{0}))

= ∑_{{β≤α \atop β(w)>0}} \binom{α − e_{w}}{β − e_{w}}(−1)^{|α−β|}f(· + h ∑_{v∈V} β(v)(v − v_{0}))

− ∑_{{β≤α \atop β(w)<α(w)}} \binom{α − e_{w}}{β}(−1)^{|α−e_{w}−β|}f(· + h ∑_{v∈V} β(v)(v − v_{0}))

= Δ_{h,v_{0}}^{α}f,

which completes the induction.

For distinct points v_{0}, v, w ∈ V, it can easily be shown that

ξ_{v}(· + h(v − v_{0})) = ξ_{v} + h, ξ_{w}(· + h(v − v_{0})) = ξ_{w},

from which it follows that, for v ≠ v_{0},

Δ_{h,v_{0}}^{e_{v}} ([ξ_{w}]_{h}^{β(w)}) = \{ \begin{matrix}β(w)h[ξ_{w}]_{h}^{β(w)−1}, & w = v; \\ 0, & w ≠ v. \end{matrix}  (1.13)

Substituting (1.13) into (1.12) then gives (1.11).

2. The diagonalisation

Our diagonalisation is based on the representation of B_{n,V} f in terms of the basis

B_{v_{0}} := {ξ^{α} : α ∈ ZZ_{+}^{V}, α(v_{0}) = 0, |α| ≤ n},  v_{0} ∈ V  (2.1)

for Π_{n}(IR^{s}). For the standard simplex and v_{0} = 0 this is the monomials. The matrix representation of B_{n,V} with respect to B_{v_{0}} is block triangular. From this we obtain the eigenvalues and a basis of eigenfunctions.

Lemma 2.2 (Block triangular form for B_{n,V}). Fix v_{0} ∈ V. Then

B_{n,V} f = ∑_{|α|≤n ; {α(v_{0})=0 \atop α∈Z Z^{V}_{+}}} \binom{n}{α}ξ^{α}Δ_{\frac{1}{n} ,v_{0}}^{α}f(v_{0}),  ∀f ∈ C(T),  (2.3)

and, in particular,

B_{n,V} (ξ^{β}) = λ_{k}^{(n)}ξ^{β} + \frac{n!}{n^{k}} ∑_{{α<β \atop α(v_{0})=0}} \frac{S(β, α)}{(n − |α|)!}ξ^{α},  ∀β ∈ ZZ_{+}^{V}, |β| ≤ n.  (2.4)

Proof: In definition (1.2), split α = γ + α(v_{0})e_{v_{0}} to obtain

B_{n,V} f = ∑_{{|γ|≤n \atop γ(v_{0})=0} ; γ∈Z Z^{V}_{+}} \binom{n}{γ}ξ^{γ}ξ_{v_{0}}^{n−|γ|}f (\frac{|γ|v_{γ} + (n − |γ|)v_{0}}{n} ) .

Since ∑_{v∈V} ξ_{v} = 1, the multinomial theorem gives

ξ_{v_{0}}^{n−|γ|} = (1 − ∑_{{v∈V \atop v≠v_{0}}} ξ_{v})^{n−|γ|} = ∑_{{|β|≤n−|γ| \atop β(v_{0})=0} ; β∈Z Z^{V}_{+}} \binom{n − |γ|}{β}(−1)^{|β|}ξ^{β}.

Hence we obtain

\begin{aligned}B_{n,V} f & = ∑_{{|γ|≤n \atop γ(v_{0})=0}} ∑_{{|β|≤n−|γ| \atop β(v_{0})=0}} \binom{n}{γ}\binom{n − |γ|}{β}(−1)^{|β|}ξ^{γ+β}f (\frac{|γ|v_{γ} + (n − |γ|)v_{0}}{n} ) \\ & = ∑_{{|α|≤n \atop α(v_{0})=0}} \binom{n}{α}ξ^{α}c_{1/n}^{α}(f), \end{aligned}

where

\binom{n}{α}c_{1/n}^{α}(f) := ∑_{γ+β=α} ∑_{{|γ|≤n \atop γ(v_{0})=0}} ∑_{{|β|≤n−|γ| \atop β(v_{0})=0}} \binom{n}{γ}\binom{n − |γ|}{β}(−1)^{|β|}f (\frac{|γ|v_{γ} + (n − |γ|)v_{0}}{n} ) .

This gives (2.3), via the simplification

c_{1/n}^{α}(f) = \frac{α!(n − |α|)!}{n!} ∑_{{γ≤α \atop γ(v_{0})=0}} \frac{n!}{γ!(n − |γ|)!} \frac{(n − |γ|)!}{(α − β)!(n − |α|)!}(−1)^{|α−γ|}

× f (v_{0} + \frac{1}{n}(|γ|v_{γ} − |γ|v_{0}))

= ∑_{{γ≤α \atop γ(v_{0})=0}} \binom{α}{γ}(−1)^{|α−γ|}f (v_{0} + \frac{1}{n} ∑_{v∈V} γ(v)(v − v_{0}))

= Δ_{\frac{1}{n} ,v_{0}}^{α}f(v_{0}).

By (1.11) and (1.7)

\begin{aligned}Δ_{h,v_{0}}^{α}ξ^{β} & = Δ_{h,v_{0}}^{α}(∑_{γ≤β} S(β, γ)h^{|β−γ|}[ξ]_{h}^{γ}) = ∑_{γ≤β} S(β, γ)h^{|β−γ|}Δ_{h,v_{0}}^{α}([ξ]_{h}^{γ}) \\ & = ∑_{α≤γ≤β} S(β, γ)h^{|β−γ|} \frac{γ!}{(γ − α)!}h^{|α|}[ξ]_{h}^{γ−α}. \end{aligned}

Since

[ξ]_{h}^{γ−α}(v_{0}) = \{ \begin{matrix}1, & γ = α; \\ 0, & otherwise, \end{matrix}

this implies

(Δ_{h,v_{0}}^{α}ξ^{β})(v_{0}) = \{ \begin{matrix}α!S(β, α)h^{|β|}, & α ≤ β; \\ 0, & otherwise. \end{matrix}  (2.5)

Hence, substituting f = ξ^{β}, |β| ≤ n, into (2.3), and using (2.5), gives

B_{n,V} (ξ^{β}) = ∑_{{|α|≤n \atop α(v_{0})=0}} \binom{n}{α}ξ^{α}(Δ_{\frac{1}{n} ,v_{0}}^{α}ξ^{β})(v_{0}) = ∑_{{α≤β \atop α(v_{0})=0}} \frac{n!}{α!(n − |α|)!}ξ^{α}α!S(β, α) (\frac{1}{n})^{|β|} ,

which can be rewritten as (2.4).

Let p_{↑} denote the leading term of the polynomial p, i.e., the unique homogeneous polynomial that satisfies

deg(p − p_{↑}) < deg(p).

Denote by Π_{k}^{0}(IR^{s}) the homogeneous polynomials of degree k. Theorem 2.6 (Diagonalisation of B_{n,V}). The multivariate Bernstein operator B_{n,V} is diagonalisable, with the same eigenvalues as the univariate Bernstein operator, i.e.,

λ_{k}^{(n)} := \frac{n!}{(n − k)!} \frac{1}{n^{k}} , k = 1, …, n, 1 = λ_{1}^{(n)} > λ_{2}^{(n)} > ⋯ > λ_{n}^{(n)} > 0.

Let P_{k,V}^{(n)} denote the λ_{k}^{(n)}–eigenspace. Then

P_{1,V}^{(n)} = Π_{1}(IR^{s}),  ∀n.  (2.7)

For k > 1, P_{k,V}^{(n)} consists of polynomials of exact degree k, which are uniquely determined by their leading term, i.e., P_{k,V}^{(n)} is isomorphic to Π_{k}^{0}(IR^{s}) via P_{k,V}^{(n)} → Π_{k}^{0}(IR^{s}) : p ↦ p_{↑}. Let p_{f}^{(n)}, deg(f) = k ≤ n, denote the λ_{k}^{(n)}–eigenfunction with leading term f_{↑}. Then (for v_{0} ∈ V fixed) a basis for P_{k,V}^{(n)} is given by

{p_{ξ^{β}}^{(n)} : β ∈ ZZ_{+}^{V}, β(v_{0}) = 0, |β| = k},  p_{ξ^{β}}^{(n)} = ∑_{α≤β} c(α, β, n)ξ^{α},  (2.8)

where the coefficients can be calculated using the recurrence formula

\begin{aligned}c(β, β, n) & := 1, \\ c(α, β, n) & := \frac{n^{|β|}}{[n − |α|]_{1}^{|β−α|} − n^{|β−α|}} ∑_{α<γ≤β} \frac{S(γ, α)}{n^{|γ|}}c(γ, β, n), α < β. \end{aligned}  (2.9)

Let M_{k,V}^{(n)} denote the dual space to P_{k,V}^{(n)}, i.e., those μ ∈ span{f ↦ f(v_{α}) : |α| = n} for which

μ(P_{j,V}^{(n)}) = {0}, ∀j ≠ k (j = 1, …, n).

The spaces P_{k}^{(n)} and M_{k}^{(n)} are S_{V} –invariant, i.e., they have the symmetry properties

S_{V} · P_{k}^{(n)} = P_{k}^{(n)}, S_{V} · M_{k}^{(n)} = M_{k}^{(n)}.

Proof: Since B_{n,V} maps onto Π_{n}(IR^{s}), it follows from (2.3) that B_{v_{0}} of (2.1) is a basis for Π_{n}(IR^{s}). We now show that the linear operator defined by ↑ : p ↦ p_{↑} takes B_{v_{0}} to another basis for Π_{n}(IR^{s}), i.e., the homogeneous polynomials

B_{v_{0}}_{↑} = ⋃_{k=0}^{n}{(ξ^{β})_{↑} : β ∈ ZZ_{+}^{V}, β(v_{0}) = 0, |β| = k}  (2.10)

are linearly independent. Suppose that

∑_{{|β|=k \atop β(v_{0})=0}} a_{β}(ξ^{β})_{↑} = 0 ⟺ ∑_{{|β|=k \atop β(v_{0})=0}} a_{β}ξ^{β} = p, ∃p ∈ Π_{k−1}(IR^{s}).

Then for δ ∈ ZZ_{+}^{V} with |δ| = k and δ(v_{0}) = 0, (1.7) and (1.11) give

Δ_{h,v_{0}}^{δ}(p) = Δ_{h,v_{0}}^{δ}( ∑_{{|β|=k \atop β(v_{0})=0}} a_{β} ∑_{α≤β} S(β, α)h^{|β−α|}[ξ]_{h}^{α}) = a_{δ} = 0,

which proves the asserted linear independence.

The linear eigenspace (2.7) is well known. For k > 1, we will use (2.3) to show that B_{n,V} has λ_{k}^{(n)}–eigenfunctions of the form

p_{ξ^{β}}^{(n)} = ∑_{α≤β} c(α, β, n)ξ^{α} = ξ^{β} + ∑_{α<β} c(α, β, n)ξ^{α} ∈ ξ^{β} + Π_{k−1}(IR^{s}) |β| = k, β(v_{0}) = 0.

Since ↑ maps these to a basis for Π_{k}^{0}(IR^{s}), they are the basis of a subspace of P_{k,V}^{(n)} which is isomorphic to Π_{k}^{0}(IR^{s}) (via p ↦ p_{↑}). The dimension count

dim(Π_{1}(IR^{s})) + ∑_{k=2}^{n} dim(Π_{k}^{0}(IR^{s})) = dim(Π_{n}(IR^{s})),

shows this subspace is all of P_{k,V}^{(n)}, and so B_{n,V} is diagonalisable.

We now show such eigenfunctions exist. Substituting

f = ∑_{α≤β} c(α, β, n)ξ^{α},  |β| = k, β(v_{0}) = 0  (2.11)

into the eigenfunction equation B_{n,V} (f) = λ_{k}^{(n)}f, and expanding using (2.4) gives

∑_{γ≤β} c(γ, β, n)B_{n,V} (ξ^{γ}) = ∑_{γ≤β} c(γ, β, n)\frac{n!}{n^{|γ|}} ∑_{α≤γ} \frac{S(γ, α)}{(n − |α|)!}ξ^{α} = λ_{k}^{(n)} ∑_{α≤β} c(α, β, n)ξ^{α}.

Equating coefficients of the linearly independent functions ξ^{α} in the above gives

\begin{aligned}λ_{k}^{(n)}c(α, β, n) & = ∑_{α≤γ≤β} \frac{n!}{n^{|γ|}} \frac{S(γ, α)}{(n − |α|)!}c(γ, β, n) \\ & = λ_{|α|}^{(n)}c(α, β, n) + ∑_{α<γ≤β} \frac{n!}{n^{|γ|}} \frac{S(γ, α)}{(n − |α|)!}c(γ, β, n). \end{aligned}  (2.12)

For α = β, (2.12) is satisfied for any choice of c(β, β, n). Suppose that c(β, β, n) := 1. For α < β, (2.12) can be rewritten as

\begin{aligned}c(α, β, n) & = \frac{1}{λ_{k}^{(n)} − λ_{|α|}^{(n)}} ∑_{α<γ≤β} \frac{n!}{n^{|γ|}} \frac{S(γ, α)}{(n − |α|)!}c(γ, β, n) \\ & = \frac{n^{|β|}}{[n − |α|]_{1}^{|β−α|} − n^{|β−α|}} ∑_{α<γ≤β} \frac{S(γ, α)}{n^{|γ|}}c(γ, β, n). \end{aligned}

This recursively defines c(α, β, n), α < β from c(β, β, n), and hence an eigenfunction of the form (2.11) exists and is given by (2.9).

For p ∈ P_{k,V}^{(n)} and A ∈ S_{V}, (1.5) gives

B_{n,V} (p ∘ A) = (B_{n,V} p) ∘ A = (λ_{k}^{(n)}p) ∘ A = λ_{k}^{(n)}(p ∘ A),

so that p ∘ A ∈ P_{k,V}^{(n)}, and P_{k,V}^{(n)} is S_{V} –invariant. Similarly, for μ ∈ M_{k,V}^{(n)} and A ∈ S_{V},

(A · μ)(P_{j,V}^{(n)}) = μ(P_{j,V}^{(n)} ∘ A) = μ(P_{j,V}^{(n)}) = {0}, j ≠ k (j = 1, …, n),

and hence M_{k,V}^{(n)} is S_{V} –invariant.

3. Elementary eigenfunctions

If p_{k}^{(n)} is the λ_{k}^{(n)}–eigenfunction (k > 1) of the univariate Bernstein operator for the standard simplex T = [0, 1], then it can be shown that the s + 1 polynomials

p_{k}^{(n)} ∘ ξ_{v}, v ∈ V

are λ_{k}^{(n)}–eigenfunctions of B_{n,V} (which are linearly independent for s > 1).

The above is a special case of the main result of this section, which effectively says that each eigenfunction of B_{n,V} is also an eigenfunction of all Bernstein operators for higher dimensional simplices when interpreted appropriately. This we describe in terms of the substitution of barycentric coordinates into so called ‘elementary’ eigenfunctions. The result is based on the following generalisation of the affine change of variables (1.5).

Let B_{n}^{IR^{d}} denote the Bernstein operator for S_{d} the standard simplex in IR^{d}, i.e.,

S_{d} := {(x_{1}, …, x_{d}) ∈ IR^{d} : x_{1}, …, x_{d} ≥ 0, x_{1} + ⋯ + x_{d} ≤ 1}.

Lemma 3.1 (B_{n,V} applied to multi–ridge functions). Let A : IR^{s} → IR^{d} be an affine map onto IR^{d}, with W := AV. Then

B_{n,V} (g ∘ A) = (B_{n,W} g) ∘ A,  ∀g ∈ C(AT).  (3.2)

In particular, for d ≤ s distinct points v_{1}, …, v_{d} ∈ V

B_{n,V} (g ∘ (ξ_{v_{1}}, …, ξ_{v_{d}})) = (B_{n}^{IR^{d}}g) ∘ (ξ_{v_{1}}, …, ξ_{v_{d}}),  ∀g ∈ C(S_{d}),  (3.3)

where ξ_{v} are the V –barycentric coordinates, and (ξ_{v_{1}}, …, ξ_{v_{d}}) is the affine map

IR^{s} → IR^{d} : x ↦ (ξ_{v_{1}}(x), …, ξ_{v_{d}}(x)).

Proof: Since A is affine, W := AV is the set of vertices of a simplex in IR^{d}, and A maps {v_{α} : α ∈ ZZ_{+}^{V}, |α| = n} onto {w_{β} : β ∈ ZZ_{+}^{W} , |β| = n}. Let ξ = (ξ_{v}) denote the V –barycentric coordinates, and η = (η_{w}) the W –barycentric coordinates. Then

B_{n,V} (g ∘ A) = ∑_{{|α|=n \atop α∈Z Z{+ \atop V}}} \binom{n}{α}ξ^{α}g(Av_{α}) = ∑_{{|β|=n \atop β∈Z Z{+ \atop W}}} ∑_{{|α|=n \atop Avα=w_{β}} ; α∈Z Z{+ \atop V}} \binom{n}{α}ξ^{α}g(w_{β}),

and

(B_{n,W} g) ∘ A = ∑_{{|β|=n \atop β∈Z Z{+ \atop W}}} \binom{n}{β}(η^{β} ∘ A) g(w_{β}).

Hence to prove (3.2), it is sufficient to show that

∑_{{|α|=n \atop Avα=w_{β}} ; α∈Z Z{+ \atop V}} \binom{n}{α}ξ^{α} = \binom{n}{β}η^{β} ∘ A,  ∀β ∈ ZZ_{W}^{+} , |β| = n.  (3.4)

We now expand the RHS of (3.4) in terms of the basis {ξ^{α} : α ∈ ZZ_{+}^{V}, |α| = n} for Π_{n}(IR^{s}). Observe that η_{w} ∘ A is the affine map IR^{s} → IR with

(η_{w} ∘ A)(v) = \{ \begin{matrix}1, & Av = w; \\ 0, & otherwise, \end{matrix} v ∈ V,

i.e.,

η_{w} ∘ A = ∑_{{v∈V \atop Av=w}} ξ_{v},

so that

η^{β} ∘ A = ( ∏_{w∈W} η_{w}^{β(w)}) ∘ A = ∏_{w∈W}(η_{w} ∘ A)^{β(w)} = ∏_{w∈W}( ∑_{{v∈V \atop Av=w}} ξ_{v})^{β(w)}.

By the multinomial theorem

( ∑_{{v∈V \atop Av=w}} ξ_{v})^{β(w)} = ( ∑_{v∈\{v:Av=w\}} ξ_{v})^{β(w)} = ∑_{{|δ|=β(w) \atop supp δ⊂\{v:Av=w\}} ; δ∈Z Z{+ \atop V}} \binom{β(w)}{δ}ξ^{δ},

and so we obtain

\binom{n}{β}η^{β} ∘ A = \binom{n}{β} ∏_{w∈W}( ∑_{{|δ|=β(w) \atop supp δ⊂\{v:Av=w\}} ; δ∈Z Z{+ \atop V}} \binom{β(w)}{δ}ξ^{δ}) = ∑_{{|α|=n \atop α∈Z Z{+ \atop V}}} c_{α} ξ^{α},  (3.5)

where the coefficients c_{α} can be determined by expanding the product. It remains to show that c_{α} equals the coefficient of ξ^{α} in LHS of (3.4), i.e.,

c_{α} = \{ \begin{matrix}( \begin{matrix}n \\ α \end{matrix} ) , & Av_{α} = w_{β}; \\ 0, & otherwise. \end{matrix}

Since V is the disjoint union ∪_{w∈W} {v : Av = w}, the coefficient c_{α} is zero unless

∑_{{v∈V \atop Av=w}} α(v) = β(w),

which implies

Av_{α} = ∑_{v∈V} \frac{α(v)}{n}Av = ∑_{w∈W} ∑_{{v∈V \atop Av=w}} \frac{α(v)}{n}w = ∑_{w∈W} \frac{β(w)}{n}w = w_{β}.

In this case, supp α ⊂ {v : Av = w}, so

c_{α} = \binom{n}{β} ∏_{w∈W} \binom{β(w)}{α|_{\{v:Av=w\}}} = \frac{n!}{β!} \frac{β!}{α!} = \binom{n}{α},

and we conclude (3.4) holds.

Since each ξ_{v_{i}} is affine, the map A := (ξ_{v_{1}}, …, ξ_{v_{d}}) is affine. From

Av = \{ \begin{matrix}e_{i}, & v = v_{i}; \\ 0, & otherwise, \end{matrix} v ∈ V,

it follows that

W := {Av : v ∈ V } = {0, e_{1}, …, e_{d}} ⟹ AT = S_{d},

i.e., B_{n,W} = B_{n}^{IR^{d}}, and (3.3) is proved.

Definition. For k > 1, the d–variate elementary λ_{k}^{(n)}–eigenfunction

p_{k_{1},...,k_{d}}^{(n)} : IR^{d} → IR, k := k_{1} + ⋯ + k_{d} ≤ n, k_{1}, …, k_{d} ≥ 1

is defined to be the λ_{k}^{(n)}–eigenfunction of B_{n}^{IR^{d}} with leading term x_{1}^{k_{1}} ⋯ x_{d}^{k_{d}}.

These can be computed via

p_{k_{1},...,k_{d}}^{(n)}(x) := ∑_{{α≤β \atop α∈Z Z^{d}_{+}}} c(α, β, n)x^{α},  β := (k_{1}, …, k_{d}),  (3.6)

where the coefficients are determined by the recurrence (2.9). This notation is consistent with that of [CW00:(2.6)], where the λ_{k}^{(n)}–eigenfunction p_{k}^{(n)} is precisely the univariate elementary eigenfunction defined above. Observe that changing the ordering of k_{1}, …, k_{d} leads to essentially the same elementary eigenfunction (the coordinates are just reordered).

Theorem 3.7 (Elementary eigenfunctions). Let v_{1}, …, v_{d} be d ≤ s+1 distinct points in V. Then

p_{k_{1},...,k_{d}}^{(n)}(ξ_{v_{1}}, …, ξ_{v_{d}}) := p_{k_{1},...,k_{d}}^{(n)} ∘ (ξ_{v_{1}}, …, ξ_{v_{d}}) : IR^{s} → IR

is the λ_{k}^{(n)}–eigenfunction of B_{n,V} with leading term (ξ_{v_{1}}^{k_{1}} ⋯ ξ_{v_{d}}^{k_{d}})_{↑}, i.e.,

p_{ξ^{β}}^{(n)} = p_{k_{1},...,k_{d}}^{(n)}(ξ_{v_{1}}, …, ξ_{v_{d}}), β := (k_{1}, …, k_{d}).

This has a factor of ξ_{v_{1}}, …, ξ_{v_{d}}. Indeed, when k_{1}, …, k_{m} > 1, k_{m+1} = ⋯ = k_{d} = 1

p_{k_{1},...,k_{d}}^{(n)}(ξ_{v_{1}}, …, ξ_{v_{d}}) = ξ_{v_{1}} ⋯ ξ_{v_{d}}g(ξ_{v_{1}}, …, ξ_{v_{m}}),  g ∈ Π_{k−d}(IR^{m}),  (3.8)

where

g(x) := g_{β̂,d}^{(n)}(x) = ∑_{{δ≤β̂ \atop δ∈Z Z^{m}_{+}}} a(δ, β̂, d, n)x^{δ},  β̂ = (k_{1} − 1, …, k_{m} − 1),  (3.9)

can be calculated from a(β̂, β̂, d, n) := 1, and for δ < β̂ the recurrence

a(δ, β̂, d, n) := \frac{n^{|β̂|}}{[n−|δ|−d]_{1}^{|β̂−δ|} − n^{|β̂−δ|}} ∑_{{δ<γ≤β̂ \atop γ∈Z Z^{m}_{+}}} \frac{S(γ+(1, …, 1), δ+(1, …, 1))}{n^{|γ|}}a(γ, β̂, d, n).

(3.10)

Moreover, all eigenfunctions of degree ≥ 2 are zero at each of the vertices V, and

p_{k−1,1}^{(n)}(x_{1}, x_{2}) = \frac{p_{k}^{(n)}(x_{1})}{x_{1} − 1}x_{2}.  (3.11)

Proof: By Lemma 3.1 and the definition of elementary eigenfunctions

B_{n,V} (p_{k_{1},...,k_{d}}^{(n)} ∘ (ξ_{v_{1}}, …, ξ_{v_{d}})) = (B_{n}^{IR^{d}}p_{k_{1},...,k_{d}}^{(n)}) ∘ (ξ_{v_{1}}, …, ξ_{v_{d}})
= (λ_{k}^{(n)}p_{k_{1},...,k_{d}}^{(n)}) ∘ (ξ_{v_{1}}, …, ξ_{v_{d}})
= λ_{k}^{(n)}p_{k_{1},...,k_{d}}^{(n)} ∘ (ξ_{v_{1}}, …, ξ_{v_{d}}),

so that p_{k_{1},...,k_{d}}^{(n)}(ξ_{v_{1}}, …, ξ_{v_{d}}) is a λ_{k}^{(n)}–eigenfunction of B_{n,V}, and since

p_{k_{1},...,k_{d}}^{(n)}(x) = x_{1}^{k_{1}} ⋯ x_{d}^{k_{d}} + lower order powers of x,

its leading term is (ξ_{v_{1}}^{k_{1}} ⋯ ξ_{v_{d}}^{k_{d}})_{↑}.

For α < (1, …, 1), i.e., α_{i} = 0 for some i, one has S(γ, α) = 0, and so by (2.9) the coefficients in (3.6) satisfy

c(α, β, n) = 0, α < (1, …, 1).

This allows us to divide (3.6) by x^{(1,...,1)} to obtain (3.8), with

g(x) = \frac{p_{k_{1},...,k_{d}}^{(n)}(x)}{x^{(1,...,1)}} = ∑_{{(1,...,1)≤α≤β \atop α∈Z Z^{d}_{+}}} c(α, β, n)x^{α−(1,...,1)}

= ∑_{{α−(1,...,1)≤(β̂,0,...,0) \atop α∈Z Z^{d}_{+}}} c(α, β, n)x^{α−(1,...,1)}

= ∑_{{δ≤β̂ \atop δ∈Z Z^{m}_{+}}} a(δ, β̂, d, n)x^{δ},

where β̂ := (k_{1} − 1, …, k_{m} − 1), i.e., β = (β̂, 0, …, 0) + (1, …, 1), and

a(δ, β̂, d, n) = c((δ, 0, …, 0) + (1, …, 1), β, n).

From the recurrence (2.9) we then obtain

a(β̂, β̂, d, n) = c(β, β, n) := 1,

and for δ < β̂

\begin{aligned}a(δ, β̂, d, n) & := \frac{n^{k}}{[n−|δ|−d]_{1}^{k−|δ|−d} − n^{k−|δ|−d}} \\ & × ∑_{(δ,0,...,0)+(1,...,1)<γ̃≤β} \frac{S(γ̃, (δ, 0, …, 0)+(1, …, 1))}{n^{|γ̃|}}c(γ̃, β, n). \end{aligned}

Making the substitution γ̃ = (γ, 0, …, 0) + (1, …, 1) in the summation above gives

\begin{aligned}∑_{{δ<γ≤β̂ \atop γ∈Z Z^{m}_{+}}} & \frac{S((γ, 0, …, 1)+(1, …, 1), (δ, 0, …, 0)+(1, …, 1))}{n^{|γ|+d}}c((γ, 0, …, 0) + (1, …, 1), β, n) \\ & = ∑_{{δ<γ≤β̂ \atop γ∈Z Z^{m}_{+}}} \frac{S(γ+(1, …, 1), δ+(1, …, 1))}{n^{|γ|+d}}a(γ, β̂, d, n), \end{aligned}

and so we obtain

\begin{aligned}a(δ, β̂, d, n) & := \frac{n^{k−d}}{[n−|δ|−d]_{1}^{k−|δ|−d} − n^{k−|δ|−d}} \\ & × ∑_{{δ<γ≤β̂ \atop γ∈Z Z^{m}_{+}}} \frac{S(γ+(1, …, 1), δ+(1, …, 1))}{n^{|γ|}}a(γ, β̂, d, n), \end{aligned}

which is (3.10).

In [CW00:(2.11)], it was shown that p_{k}^{(n)}(1) = 0, k ≥ 2. This together with the fact ξ_{v}(w) := \{ \begin{matrix}1, & w = v; \\ 0, & otherwise \end{matrix}

implies all the eigenfunctions (3.8) of degree ≥ 2 are zero at the vertices v ∈ V.

Example 1. For p_{2,1,...,1}^{(n)} we have β̂ = 1 ∈ ZZ (d = k − 1), and g is the univariate linear polynomial g(x) = x + a(0, 1, k − 1, n), where

a(0, 1, k − 1, n) := \frac{n^{1}}{[n − (k − 1)]_{1}^{1} − n^{1}} \frac{S((2, 1, …, 1), (1, …, 1))}{n^{1}} = −\frac{1}{(k − 1)}.

Thus

p_{2,1,...,1}^{(n)}(ξ_{v_{1}}, …, ξ_{v_{k−1}}) = ξ_{v_{1}} ⋯ ξ_{v_{k−1}} (ξ_{v_{1}} − \frac{1}{(k − 1)})

is a λ_{k}^{(n)}–eigenfunction of B_{n,V} whenever 2 ≤ k ≤ s + 2.

Since this function is independent of n, it follows that there are eigenfuctions of degree k which are shared by all B_{n}, k ≥ n, for sufficiently large s.

Example 2. For p_{3,1,...,1}^{(n)} we have β̂ = 2 (d = k − 2), and g is the univariate quadratic polynomial g(x) = x^{2} + a(1, 2, k − 2, n)x + a(0, 2, k − 2, n), where

a(1, 2, k − 2, n) := \frac{n^{2}}{[n − 1 − (k − 2)]_{1}^{1} − n} \frac{S(3, 2)}{n^{2}} = −\frac{3}{(k − 1)},

and

a(0, 2, k − 2, n) := \frac{n^{2}}{[n − (k − 2)]_{1}^{2} − n^{2}} \{\frac{S(2, 1)}{n} \frac{−3}{(k − 1)} + \frac{S(3, 1)}{n^{2}}\}

= \frac{n^{2}}{−2nk + 3n + k^{2} − 3k + 2} \{\frac{−3}{n(k − 1)} + \frac{1}{n^{2}} \}

= \frac{3n − k + 1}{(2nk − 3n − k^{2} + 3k − 2)(k − 1)}.

Thus

ξ_{v_{1}} ⋯ ξ_{v_{k−2}} (ξ_{v_{1}}^{2} − \frac{3}{(k − 1)}ξ_{v_{1}} + \frac{3n − k + 1}{(2nk − 3n − k^{2} + 3k − 2)(k − 1)})

is a λ_{k}^{(n)}–eigenfunction of B_{n,V} whenever 3 ≤ k ≤ s + 3.

Example 3. For p_{2,2,1,...,1}^{(n)} we have β̂ = (1, 1) (d = k − 2), and g is the bivariate quadratic polynomial g(x) = x_{1}x_{2} + c_{1}(x_{1} + x_{2}) + c_{2}, where

\begin{aligned}c_{1} & = a((1, 0), β̂, k − 2, n) = a((0, 1), β̂, k − 2, n) \\ & = \frac{n^{2}}{(n − k + 1) − n} \frac{S(2, 1)}{n^{2}} = −\frac{1}{(k − 1)}, \end{aligned}

and

\begin{aligned}c_{2} = a((0, 0), β̂, k − 2, n) & = \frac{n^{2}}{[n − (k − 2)]_{1}^{2} − n^{2}} \{2\frac{S(2, 1)}{n} \frac{−1}{(k − 1)} + \frac{S((2, 2), (1, 1))}{n^{2}}\} \\ & = \frac{2n − k + 1}{(2nk − 3n − k^{2} + 3k − 2)(k − 1)}. \end{aligned}

Thus, for k ≥ 4,

p_{2,2,1,...,1}^{(n)}(x) = x_{1} ⋯ x_{k−2} (x_{1}x_{2} − \frac{x_{1} + x_{2}}{(k − 1)} + \frac{2n − k + 1}{(2nk − 3n − k^{2} + 3k − 2)(k − 1)}) .

A list of the elementary eigenfunctions up to degree 5 is provided in the appendix.

4. Limiting eigenfunctions

Here we show that the λ_{k}^{(n)}–eigenfunctions p_{f}^{(n)} converge as n → ∞ to a limit p_{f}^{∗}. Moreover, the limit of the factor (3.9) of an elementary eigenfunction is a multivariate Jacobi polynomial. This extends Theorems 4.1 and 4.5 of [CW00] to the multivariate setting. Let e_{i} be the i–th unit vector in IR^{m}, and (β)_{α} the multivariate shifted factorial

(β)_{α} := (β_{1})_{α_{1}} ⋯ (β_{d})_{α_{d}},  (β_{i})_{α_{i}} := β_{i}(β_{i} + 1) ⋯ (β_{i} + α_{i} − 1),  β ∈ IR^{d}, α ∈ ZZ_{+}^{d},

with (2)_{δ} := (2)_{δ_{1}} ⋯ (2)_{δ_{m}}.

Theorem 4.1 (Limiting eigenfunctions). Express ξ^{β}, β ∈ ZZ_{+}^{V}, |β| = k in the form

ξ^{β} = ξ_{v_{1}}^{k_{1}} ⋯ ξ_{v_{d}}^{k_{d}},  k = k_{1} + ⋯ + k_{d},  k_{1}, …, k_{m} > 1, k_{m+1} = ⋯ = k_{d} = 1,

where v_{1}, …, v_{d} are d ≤ s + 1 distinct points in V. Then the coefficients of (3.9) satisfy

lim_{n→∞} a(δ, β̂, d, n) = a^{∗}(δ, β̂, d) := (−1)^{k−d} \frac{k_{1}! ⋯ k_{m}!}{(k + d − 1)_{k−d}} (k + d − 1)_{|δ|} \frac{(−β̂)_{δ}}{(2)_{δ}} \frac{1}{δ!}. (4.2)

Thus, p_{ξ^{β}}^{(n)} converges uniformly on T to p_{ξ^{β}}^{∗} = ξ_{v_{1}} ⋯ ξ_{v_{d}}g(ξ_{v_{1}}, …, ξ_{v_{m}}), where

g(x) := g_{β̂,d}^{∗}(x) := ∑_{{δ≤β̂ \atop δ∈Z Z^{m}_{+}}} a^{∗}(δ, β̂, d) x^{δ},  β̂ := (k_{1} − 1, …, k_{m} − 1).  (4.3)

Proof: First we prove by strong induction on |β̂ − δ| that a(δ, β̂, d, n) converges to a limit a^{∗}(δ, β̂, d) as n → ∞, which satisfies the recurrence

a^{∗}(δ, β̂, d) = \frac{−1}{|β̂ − δ|(k + d + |δ| − 1)} ∑_{{i=1 \atop δ_{i}+1≤k_{i}−1}}^{m}(δ_{i} + 1)(δ_{i} + 2)a^{∗}(δ + e_{i}, β̂, d), δ < β̂.

(4.4)

Clearly lim_{n→∞} a(β̂, β̂, d, n) = 1, which begins the induction. Suppose δ < β̂. Since

[n−|δ|−d]_{1}^{|β̂−δ|} − n^{|β̂−δ|} = \frac{1}{2}|β̂ − δ|(1 − k − d − |δ|)n^{|β̂−δ|−1} + lower order powers of n,

all the coefficients

\frac{n^{|β̂−γ|}}{[n−|δ|−d]_{1}^{|β̂−δ|} − n^{|β̂−δ|}} S(γ+(1, …, 1), δ+(1, …, 1)),

of a(γ, β̂, d, n) in (3.10) converge to 0 as n → ∞, except those for γ = δ + e_{i} ≤ β̂ which converge to

\frac{S(δ + e_{i} + (1, …, 1), δ + (1, …, 1))}{\frac{1}{2}|β̂ − δ|(1 − k − d − |δ|)} = −\frac{(δ_{i} + 1)(δ_{i} + 2)}{|β̂ − δ|(k + d + |δ| − 1)}.

Using this and the inductive hypothesis, we can take the limit of (3.10) to obtain

lim_{n→∞} a(δ, β̂, d, n) = a^{∗}(δ, β̂, d),

which satisfies (4.4).

The limits a^{∗}(δ, β̂, d) are uniquely determined by a^{∗}(β̂, β̂, d) = 1 and (4.4). We now show that the a^{∗} defined in (4.2) satisfies these, and so gives the desired limits. The case δ = β̂ is trivial, and so it suffices to show

b(δ, β̂, d) := (k + d − 1)_{|δ|} \frac{(−β̂)_{δ}}{(2)_{δ}} \frac{1}{δ!}

satisfies the recurrence (4.4). For δ < β̂, we compute

∑_{{i=1 \atop δ_{i}+1≤k_{i}−1}}^{m}(δ_{i} + 1)(δ_{i} + 2)b(δ + e_{i}, β̂, d)

= ∑_{{i=1 \atop δ_{i}+1≤k_{i}−1}}^{m}(δ_{i} + 1)(δ_{i} + 2)(k + d − 1)_{|δ+e_{i}|} \frac{(−β̂)_{δ+e_{i}}}{(ˆ2)_{δ+e_{i}}} \frac{1}{(δ + e_{i})!}

= −(k + d − 1)_{|δ|} \frac{(−β̂)_{δ}}{(ˆ2)_{δ}} \frac{1}{δ!} (k + d + |δ| − 1) ∑_{{i=1 \atop δ_{i}+1≤β̂_{i}}}^{m} β̂_{i} − δ_{i}

= −b(δ, β̂, d) (k + d + |δ| − 1)|β̂ − δ|,

as required. Since the sequence p_{ξ^{β}}^{(n)} is contained in the finite dimensional space Π_{k}, it converges to p_{ξ^{β}}^{∗} ∈ Π_{k} in any norm, and in particular uniformly.

Let P_{k,V}^{∗} denote the space of limiting eigenfunctions, i.e.,

P_{1,V}^{∗} := Π_{1}(IR^{s}), P_{k,V}^{∗} := span{p_{ξ^{β}}^{∗} : β ∈ ZZ_{+}^{V}, β(v_{0}) = 0, |β| = k}, k > 1,

which is S_{V} –invariant. It follows immediately that each sequence of eigenfunctions p_{f}^{(n)}, f_{↑} ∈ Π_{k}^{0}(IR^{s}) converges as n → ∞ to some p_{f}^{∗} ∈ P_{k,V}^{∗}.

The Lauricella function F = F_{A} (see, e.g., [E76:Chap. 2]) with arguments c a scalar, and β, γ, x vectors from IR^{d} (or IR^{V}) is defined by

F (c, β; γ; x) := ∑_{α∈ZZ_{+}^{d}}(c)_{|α|} \frac{(β)_{α}}{(γ)_{α}} \frac{x^{α}}{α!} , c ∈ IR, β, γ, x ∈ IR^{d}.

Theorem 4.5 (Identification of p_{ξ^{β}}^{∗}). The function g of (4.3) can be expressed as

g_{β̂,d}^{∗}(x) = (−1)^{k−d} \frac{k_{1}! ⋯ k_{m}!}{(k + d − 1)_{k−d}} F (k + d − 1, −β̂; ˆ2; x),  ˆ2 := (2, …, 2).

Proof: From (4.2), (4.3) and the definition of F, we have

\begin{aligned}g_{β̂,d}^{∗}(x) & = (−1)^{k−d} \frac{k_{1}! ⋯ k_{m}!}{(k + d − 1)_{k−d}} ∑_{{δ≤β̂ \atop δ∈Z Z^{m}_{+}}} (k + d − 1)_{|δ|} \frac{(−β̂)_{δ}}{(2)_{δ}} \frac{x^{δ}}{δ!} \\ & = (−1)^{k−d} \frac{k_{1}! ⋯ k_{m}!}{(k + d − 1)_{k−d}} F (k + d − 1, −β̂; ˆ2; x). \end{aligned}

Define d–vectors β := (β̂, 0, …, 0), |β| = k −d, κ := (1, …, 1) and ξ = (ξ_{v_{1}}, …, ξ_{v_{d}}). Then

F (k + d − 1, −β̂;ˆ;(ξ_{v_{1}}, …, ξ_{v_{m}})) = F (|β| + |κ| + (d − 1), −β, κ + 1, ξ).

In [W01] it is shown that the factor g(ξ_{v_{1}}, …, ξ_{v_{m}}), d ≥ 2 of p_{ξ^{β}}^{∗} is the (multivariate) Jacobi polynomial of degree k − d for the simplex with vertices {v_{1}, …, v_{d}} and weight ξ_{v_{1}} ⋯ ξ_{v_{d}} which has leading term (ξ_{v_{1}}^{k_{1}−1} ⋯ ξ_{v_{m}}^{k_{m}−1})_{↑}.

Example 1. Consider the univariate case T = S_{1} := [0, 1]. Here the barycentric coordinates are ξ_{0}(x) = 1 − x and ξ_{1}(x) = x. For ξ^{β}(x) = x^{k} we have d = m = 1, β̂ = k − 1, giving

\begin{aligned}p_{k}^{∗}(x) & = x(−1)^{k−1} \frac{k!}{(k)_{k−1}} F (k, 1 − k; 2; x) \\ & = x(−1)^{k−1} \frac{k!(k − 1)!}{(2k − 2)!}_{2}F_{1}(1 − k, k; 2; x). \end{aligned}

Similarly, the leading term of x^{k−1}(1 − x) is −x^{k}. So taking k_{1} = k − 1, k_{2} = 1, m = 1, d = 2, β̂ = k − 2 gives

\begin{aligned}p_{k}^{∗}(x) & = −x(1 − x)(−1)^{k−2} \frac{(k − 1)!}{(k + 1)_{k−2}} F (k + 1, 2 − k; 2; x) \\ & = x(x − 1)(−1)^{k} \frac{k!(k − 1)!}{(2k − 2)!}_{2}F_{1}(2 − k, k + 1; 2; x), \end{aligned}

and so we have the result of [CW00:Th. 4.5], that

p_{k}^{∗}(x) = \frac{k!(k − 2)!}{(2k − 2)!}x(x − 1)P_{k−2}^{(1,1)}(2x − 1),  k ≥ 2,  (4.6)

where P_{j}^{(1,1)} are the (univariate) Jacobi polynomials which are orthogonal with respect to the weight (1 − t)(1 + t) on the interval t ∈ [−1, 1].

Example 2. For each p_{k_{1},...,k_{d}}, d ≥ 2,

(x_{1}, …, x_{d−1}) ↦ \frac{p_{k_{1},...,k_{d}}^{∗}(x_{1}, …, x_{d−1}, 1 − x_{1} − ⋯ − x_{d−1})}{x_{1} ⋯ x_{d−1}(1 − x_{1} − ⋯ − x_{d−1})}

is a Jacobi polynomial of degree k −d for S_{d−1} with weight x_{1} ⋯ x_{d−1}(1−x_{1} −⋯−x_{d−1}).

5. B_{n,V} applied to shifted factorials

The following result is of independent interest. In particular, it shows that

B_{n,V} ([ξ]_{1/n}^{β}) = λ_{k}^{(n)}ξ^{β},  |β| = k ≤ n,  (5.1)

which can be used to give an alternative proof of the diagonalisation of B_{n,V}.

Theorem 5.2 (B_{n,V} applied to shifted factorials). Recall for β ∈ ZZ_{+}^{V} with |β| = k ≤ n,

[ξ]_{1/n}^{β} := ∏_{v∈V} ξ_{v} (ξ_{v} − \frac{1}{n}) (ξ_{v} − \frac{2}{n}) ⋯ (ξ_{v} − \frac{β(v) − 1}{n}) ∈ Π_{k}.

Then

B_{n,V} ([ξ]_{1/n}^{β}g) = λ_{k}^{(n)}ξ^{β}B_{n−k,V} (g (\frac{n−k}{n} · −\frac{k}{n}v_{β})),  ∀g ∈ C(T),  (5.3)

where the Bernstein polynomials in (5.3) depend only on the values

{g(v_{α}) : α ∈ ZZ_{+}^{V}, |α| = n, α ≥ β}.  (5.4)

In particular, taking g = 1 in (5.3) gives (5.1). It can also be shown that

B_{n−k,V} (g (\frac{n−k}{n} · −\frac{k}{n}v_{β})) = (B_{n−k,W} g) ∘ (\frac{n−k}{n} · −\frac{k}{n}v_{β}) ,  W := \frac{n−k}{n} V −\frac{k}{n}v_{β}. (5.5)

Proof: Since v_{α} is an affine combination of the points in V

ξ_{v}(v_{α}) = ∑_{w∈V} \frac{α(w)}{n}ξ_{v}(w) = \frac{α(v)}{n}, α ∈ ZZ_{+}^{V}, |α| = n,

and we have

([ξ]_{1/n}^{β})(v_{α}) = ∏_{v∈V} \frac{α(v)}{n} (\frac{α(v) − 1}{n}) (\frac{α(v) − 2}{n}) ⋯ (\frac{α(v) − (β(v) − 1)}{n})

= \frac{1}{n^{k}} \{ \begin{matrix}α!/(α − β)!, & α ≥ β; \\ 0, & otherwise. \end{matrix}

(5.6)

This implies B_{n,V} ([ξ]_{1/n}^{β}g) depends only on (5.4). For α ≥ β, |α| = n

v_{α} = ∑_{v∈V} \frac{α(v)v}{n} = \frac{n − k}{n} ∑_{v∈V} \frac{(α(v) − β(v))v}{n − k} + \frac{k}{n} ∑_{v∈V} \frac{β(v)v}{k} = \frac{n − k}{n} v_{α−β} + \frac{k}{n}v_{β},

and hence by (5.6) we obtain

B_{n,V} ([ξ]_{1/n}^{β}g) = \frac{1}{n^{k}} ∑_{{|α|=n \atop α≥β}} \binom{n}{α}ξ^{α} \frac{α!}{(α − β)!}g(v_{α})

= \frac{n!}{(n − k)!} \frac{1}{n^{k}} ξ^{β} ∑_{{|α−β|=n−k \atop α−β≥0}} \binom{n − k}{α − β}ξ^{α−β}g (\frac{n−k}{n} v_{α−β}−\frac{k}{n}v_{β})

= λ_{k}^{(n)}ξ^{β}B_{n−k,V} (g (\frac{n−k}{n} · −\frac{k}{n}v_{β})).

Applying (3.2) with A := \frac{n−k}{n} · −\frac{k}{n}v_{β}, gives (5.5).

Let V_{n} := {v_{α} : α ∈ ZZ_{+}^{V}, |α| = n}, then Theorem 5.2 relates the support of the mesh function f|_{V_{n}} to factors of B_{n}(f). Corollary 5.7. Choose β ∈ ZZ_{+}^{V} with |β| = k ≤ n, and define

V_{β} := {v_{α} ∈ V_{n} : α ≥ β}.

Then the following are equivalent

supp(f|_{V_{n}}) ⊂ V_{β} ⟺ [ξ]_{1/n}^{β} | (f|_{V_{n}}) ⟺ ξ^{β} | B_{n,V} (f), ∀f ∈ C(T).

When this holds

B_{n}(f) = λ_{k}^{(n)}ξ^{β}(B_{n−k,W} (f/[ξ]_{1/n}^{β})) ∘ (\frac{n−k}{n} · −\frac{k}{n}v_{β}) , W := \frac{n−k}{n} V − \frac{k}{n}v_{β}.

Proof: From (5.6), we have

V_{β} = {v_{α} ∈ V_{n} : [ξ]_{1/n}^{β}(v_{α}) ≠ 0},

and so

[ξ]_{1/n}^{β} | (f|_{V_{n}}) ⟺ f(v_{α}) = 0, ∀v_{α} ∈ V_{n} \ V_{β} ⟺ supp(f|_{V_{n}}) ⊂ V_{β}.

By Theorem 5.2,

[ξ]_{1/n}^{β} | (f|_{V_{n}}) ⟺ f|_{V_{n}} = ([ξ]_{1/n}^{β})|_{V_{n}}(f/[ξ]_{1/n}^{β})|_{V_{n}}
⟺ B_{n,V} (f) = B_{n,V} ([ξ]_{1/n}^{β}(f/[ξ]_{1/n}^{β}))
⟺ B_{n,V} (f) = λ_{k}^{(n)}ξ^{β}(B_{n−k,W} (f/[ξ]_{1/n}^{β})) ∘ (\frac{n−k}{n} · −\frac{k}{n}v_{β}) .

Now suppose that ξ^{β} | B_{n,V} (f), then

\begin{aligned}B_{n,V} (f)/ξ^{β} = ∑_{|α|=n} \binom{n}{α}ξ^{α−β}f(v_{α}) ∈ Π_{n−k}(IR^{s}) & ⟹ f(v_{α}) = 0, ∀α − β ≱ 0 \\ & ⟹ supp f|_{V_{n}} ⊂ V_{β}. \end{aligned}

6. Concluding remarks

We conclude with some comments about those parts of [CW00] which have not been generalised here.

The common zeroes of the eigenspaces P_{k,V}^{(n)} do not have the rich structure of the univariate situation (k real zeros in [0, 1] with estimates on their location). Indeed, by considering the factored form of elementary eigenfunction p_{k−s−1,1,...,1}^{(n)} it follows that the common zeros are just the vertices V.

By choosing a basis of eigenfunctions for B_{n,V}, say that of (2.8) with p_{α}^{(n)} := p_{ξ^{α}}^{(n)}, one can write down a diagonal form

B_{n,V} f = ∑_{k=0}^{n} λ_{k}^{(n)} ∑_{{|α|=k \atop α(v_{0})=0}} p_{α}^{(n)}μ_{α}^{(n)}(f),  ∀f ∈ C(T),  (6.1)

where the dual functionals μ_{α}^{(n)} ∈ M_{k,V}^{(n)} can be found explicitly by solving the linear system obtained from

μ_{α}^{(n)}(p_{β}^{(n)}) = δ_{α,β}, ∀α, β.

None of the formulæ so obtained are nice enough to be worth recording. Recent results of [WX01] using (tight) frames to represent Jacobi polynomials on a simplex indicate that it might be more profitable to consider a redundant, but more symmetric, representation of the form

B_{n,V} f = ∑_{k=0}^{n} λ_{k}^{(n)} ∑_{|α|=k} p_{α}^{(n)}μ_{α}^{(n)}(f), ∀f ∈ C(T),

where the inner sum involves all of the S_{V} –invariant spanning set {p_{α}^{(n)} : α ∈ ZZ_{+}^{V}, |α| = k}.

In [CW00] it was shown that dual functionals such as μ_{α}^{(n)} in (6.1) have a limit as n → ∞ (as functionals on the polynomials). The argument given relied on the fact that dividing p_{k}^{(n)}, k ≥ 2 by the product of the barycentric coordinates (for the interval) gave a sequence of Jacobi polynomials for which an orthogonal expansion could be used. In the multivariate case this is no longer possible (not all eigenfunctions are divisable by each barycentric coordinate). It is believed that such limits do exist, and that they might be found by an appropriate orthogonal expansion (possibly involving Sobolev orthogonality).

There has been some work on iterates of the bivariate Bernstein operator by [LiP87] and [CF93] generalising the methods of [KR67] (see the comments in [CW00:Sect. 5]). By setting the eigenvalues in (6.1) to 1 we obtain, similarly to the univariate case, the operator

L_{n,V} f = ∑_{k=0}^{n} ∑_{{|α|=k \atop α(v_{0})=0}} p_{α}^{(n)}μ_{α}^{(n)}(f), ∀f ∈ C(T)

of Lagrange interpolation from Π_{n} at the ‘simplex points’ {v_{α} : |α| = n}. The classes of Bernstein quasi–interpolant operators proposed in [CW00] can then be defined in the obvious way. The eigenstructure of the multivariate Kantorovich operator can be deduced in the same was as in the univariate case, see, e.g., [LiS96] (and references therein) for a discussion of the properties of this operator.

7. Appendix

List of the elementary eigenfunctions for k = 2, …, 5

degree 2, i.e., λ_{2}^{(n)} = \frac{n!}{(n − 2)!n^{2}}

p_{2}^{(n)}(x) = x_{1}(x_{1} − 1)
p_{1,1}^{(n)}(x) = x_{1}x_{2}

degree 3, i.e., λ_{3}^{(n)} = \frac{n!}{(n − 3)!n^{3}}

p_{3}^{(n)}(x) = x_{1}(x_{1} − 1/2)(x_{1} − 1)
p_{2,1}^{(n)}(x) = x_{1}x_{2}(x_{1} − 1/2)
p_{1,1,1}^{(n)}(x) = x_{1}x_{2}x_{3}

degree 4, i.e., λ_{4}^{(n)} = \frac{n!}{(n − 4)!n^{4}}

p_{4}^{(n)}(x) = x_{1}(x_{1} − 1) (x_{1}^{2} − x_{1} + \frac{n − 1}{5n − 6})
p_{3,1}^{(n)}(x) = x_{1}x_{2} (x_{1}^{2} − x_{1} + \frac{n − 1}{5n − 6})
p_{2,2}^{(n)}(x) = x_{1}x_{2} (x_{1}x_{2} − \frac{1}{3}x_{1} − \frac{1}{3}x_{2} + \frac{2n − 3}{3(5n − 6)})
p_{2,1,1}^{(n)}(x) = x_{1}x_{2}x_{3}(x_{1} − 1/3)
p_{1,1,1,1}^{(n)}(x) = x_{1}x_{2}x_{3}x_{4}

degree 5, i.e., λ_{5}^{(n)} = \frac{n!}{(n − 5)!n^{5}}

p_{5}^{(n)}(x) = x_{1}(x_{1} − 1/2)(x_{1} − 1) (x_{1}^{2} − x_{1} + \frac{n − 1}{7n − 12})
p_{4,1}^{(n)}(x) = x_{1}x_{2}(x_{1} − 1/2) (x_{1}^{2} − x_{1} + \frac{n − 1}{7n − 12})
p_{3,2}^{(n)}(x) = x_{1}x_{2}(x_{1}^{2}x_{2} − \frac{1}{4}x_{1}^{2} − \frac{3}{4}x_{1}x_{2} + \frac{3(n − 2)}{2(7n − 12)}x_{1}

+ \frac{3n − 4}{4(7n − 12)}x_{2} − \frac{n − 2}{4(7n − 12)})

p_{3,1,1}^{(n)}(x) = x_{1}x_{2}x_{3} (x_{1}^{2} − \frac{3}{4}x_{1} + \frac{3n − 4}{4(7n − 12)})
p_{2,2,1}^{(n)}(x) = x_{1}x_{2}x_{3} (x_{1}x_{2} − \frac{1}{4}x_{1} − \frac{1}{4}x_{2} + \frac{n − 2}{2(7n − 12)})
p_{2,1,1,1}^{(n)}(x) = x_{1}x_{2}x_{3}x_{4}(x_{1} − 1/4)
p_{1,1,1,1,1}^{(n)}(x) = x_{1}x_{2}x_{3}x_{4}x_{5}

Formulæ for the elementary eigenfunctions of degree k

\begin{aligned}p_{1,...,1}^{(n)}(x) & = x_{1} ⋯ x_{k} \\ p_{2,1,...,1}^{(n)}(x) & = x_{1} ⋯ x_{k−1} (x_{1} − \frac{1}{(k − 1)}) \\ p_{3,1,...,1}^{(n)}(x) & = x_{1} ⋯ x_{k−2} (x_{1}^{2} − \frac{3}{(k − 1)}x_{1} + \frac{3n − k + 1}{(2nk − 3n − k^{2} + 3k − 2)(k − 1)}) \\ p_{2,2,1,...,1}^{(n)}(x) & = x_{1} ⋯ x_{k−2} (x_{1}x_{2} − \frac{x_{1} + x_{2}}{(k − 1)} + \frac{2n − k + 1}{(2nk − 3n − k^{2} + 3k − 2)(k − 1)}) \\ p_{k−1,1}^{(n)}(x) & = \frac{p_{k}^{(n)}(x_{1})}{x_{1} − 1}x_{2} \end{aligned}

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