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Ideal interpolation
Carl de Boor
Abstract. A linear interpolation scheme is termed ‘ideal’ when its errors form a polynomial ideal. The paper surveys basic facts about ideal interpolation and raises some questions.
Introduction
Ideal interpolation is, by definition, given by a linear projector on the space Π of polynomials whose kernel is a polynomial ideal. It is therefore also any linear map, as used in algebra, that associates a polynomial with its normal form with respect to a polynomial ideal. This article lists basic facts about ideal interpolation and raises some questions.
Definition and basic algebraic facts
If P is a linear projector of finite rank on the linear space X over the commutative field IF with algebraic dual X′, then we can think of it as providing a linear interpolation scheme on X: For each g ∈ X, f = P g is the unique element of ran P := P (X) for which
λf = λg, ∀λ ∈ ran P ′^ = {λ ∈ X′^ : λP = λ},
with P ′^ the dual of P , i.e., the linear map X′^ → X′^ : λ 7 → λP. In other words, given that ker P := {g ∈ X : P g = 0} = ran(id − P ),
ran P ′^ = (ker P )⊥^ := {λ ∈ X′^ : ker P ⊂ ker λ},
the set of interpolation conditions matched by P. Not surprisingly, there are exactly as many independent conditions as there are degrees of freedom, i.e.,
dim ran P = dim ran P ′.
Put into more practical terms, if the column maps
V : IFn^ → X : a 7 →
∑^ n
j=
vj a(j) =: [v 1 ,... , vn]a
and
Λ : IFn^ → X′^ : a 7 →
∑^ n
j=
λj a(j) =: [λ 1 ,... , λn]a,
into X and X′^ respectively, are such that their Gram matrix
ΛtV := (λivj : i, j = 1:n)
is invertible, then, in particular, both V and Λ are 1-1, hence bases for their respective ranges and there is, for given b ∈ IFn, exactly one element, call it V a, of ran V that satisfies the equation
Λt(V a) = b,
thus giving rise to the map P = V (ΛtV )−^1 Λt
on X, evidently a linear projector, that associates g ∈ X with the unique element f = P g in ran V = ran P for which Λtf := (λif : i = 1:n)
agrees with Λtg, hence λf = λg for all λ ∈ ran Λ = ran P ′. Consider now, in particular, the linear space
Π = Π(IFd)
of all IF-valued polynomials in d real (IF = IR) or complex (IF = C) variables. It will be important that Π is also a ring under pointwise multiplication,
(pq)(x) := p(x)q(x), p, q ∈ Π, x ∈ Cd.
In [Bi79], Garrett Birkhoff defined ideal interpolation as a linear projector P on Π whose nullspace or kernel is an ideal, i.e., not only closed under addition and multiplication by scalars but also under (pointwise) multiplication by arbitrary polynomials. Lagrange interpolation is mentioned by Birkhoff as a particular example. However, ideal projectors are already looked at carefully in [M76], where they are called ‘Hermite interpolation’. Ideal projectors are, in a sense, aware of the multiplicative structure of Π, hence we would expect insights from considering their interaction with multiplication, as exhibited by the following very handy fact.
Lemma 1 ([dB03]). A linear projector P on Π is ideal if and only if
(2) P (pq) = P (pP q), ∀p, q ∈ Π.
Proof: The condition (2) is equivalent to having
P (Π(id − P )(Π)) = { 0 },
and, since P is a linear projector hence (id − P )(Π) = ker P , this is equivalent to
Π ker P ⊂ ker P,
hence, given that ker P is a linear subspace, to ker P being an ideal.
An ideal projector is completely determined by its action on a subspace only slightly larger than its range. This is readily seen by the following considerations. Each ideal projector P induces a map,
(3) M : Π → L(ran P ) : p 7 → Mp,
on Π into the space L(ran P ) of linear maps on ran P , by the prescription
(4) Mp : ran P → ran P : f 7 → P (pf ), p ∈ Π.
Indeed, Mp so defined is a linear map on ran P , and depends linearly on p, hence the map M is well-defined and is linear. More than that, for arbitrary p, q ∈ Π and f ∈ ran P ,
MqMpf − Mqpf = P (qP (pf )) − P (qpf ) = 0,
the last equality by (2), hence M is also a homomorphism, on the ring Π into L(ran P ) considered as a ring with respect to map composition as multiplication. Also, since Π is a commutative ring, so is ran M , even though it is a subring of the noncommutative ring L(ran P ).
Example As an example, consider the following situation, discussed in [Sh] in the bivariate case: P is an ideal projector with range F := ran[()j 1 : j = 0:n−1],
and IF = C hence ()n 1 − P ()n 1 , considered as a univariate polynomial, has n zeros counting multiplicities. Assume, finally, that these zeros are all simple, hence
(()n 1 − P ()n 1 )(x) =:
∏^ n
j=
(x(1) − τ (j))
defines the sequence τ with pairwise distinct entries. Set
zj := (τj , (P () 2 )(τj ),... , (P ()d)(τj )), j = 1:n.
Then any p ∈ F vanishing on z is necessarily zero, hence since z has n entries and dim F = n, there is, for each p ∈ Π, exactly one element of F , call it Rp, that agrees with p on z. I claim that R = P and, by Proposition 7, need to check this only for ()α^ with α(1) < n, α(2:d) = (δij : j = 2:d), i = 2:d, since it is already evident for α = (n, 0 ,... , 0), hence for α = (m, 0 ,... , 0) for all m ∈ IN, by Proposition 7 (since ()n 1 spans an algebraic complement of F in Π 1 (F ) when considering only the ring of univariate polynomials). For the check, notice that
(R()i)(zj ) = ()i(zj ) = zj (i) = (P ()i)(τj ) = (P ()i)(zj ),
hence R = P on ()i for i = 2:d. With that, for any j,
P (()j 1 ()i) = P (()j 1 P ()i) = R(()j 1 R()i) = R(()j 1 ()i),
the middle equality since P ()i = R()i ∈ F , while the other two equalities follow from P and R being ideal.
A basis for the ideal ker P
By (6), ker M ⊂ ker P , while, if p ∈ ker P , then p(M )f = P (pf ) = P (f P p) = P 0 = 0 for all f in ran P which is the domain of p(M ), hence then p(M ) = 0. Thus, altogether,
(8) ker M = ker P.
Hence, by Proposition 7, we should be able to derive ker P from () 0 − P () 0 and the action of the restriction
N := P (^) Π 1 (F )
of P to Π 1 (F ), with F := ran P.
Proposition 9. If () 0 ∈ ran P , then
(10) ker P = ideal(ker N ) =: I.
Proof: Since ker N = ker P ∩ Π 1 (F ) and ker P is an ideal, we immediately have
ker P ⊇ I.
For the converse containment, let
Πk(S) :=
|α|≤k
()αS, ∅ 6 = S ⊂ Π.
Then, for any additive subset S of Π, we have
Πr+s(S) = Πr (Πs(S)).
In particular, Πk := Πk(IF) = Π 1 (Πk− 1 ).
Specifically, ∪k Πk(F ) = Π since we assumed F = ran P to contain () 0. Therefore, we know that ker P ⊆ I once we show, by induction on k, that
p ∈ ker P ∩ Πk (F ) =⇒ p ∈ I.
For k = 1, this is so by definition of I. So assuming it to hold for all k < h, let p ∈ ker P ∩ Πh(F ). Then
p =
j=0:d
()j pj
with pj ∈ Π<h(F ), hence (id − P )pj is in Π<h(F ) + F = Π<h(F ) as well as in ker P , hence in I by induction hypothesis. Thus,
p ∈
j
()j (P pj + I) =
j
()j P pj + I,
while, by (2), P
j ()j^ P pj^ =^ P^
j ()j^ pj^ =^ P p^ = 0, hence^
j ()j^ P pj^ ∈^ ker^ P^ ∩^ Π^1 (F^ ), therefore in^ I. It follows that ker P is generated, as an ideal, by any (vector-space) basis for ker P ∩ Π 1 (F ). Further, such a basis is readily obtained in the form
(b − N b : b ∈ B),
with B any basis for an algebraic complement of F in Π 1 (F ). As the example of bivariate tensor-product interpolation to gridded data shows, the resulting (ideal) basis may be far from minimal.
Mourrain’s condition
Proposition 9 (though not the proof given here) is essentially due to Mourrain [Mo] who proved it under the additional assumption that F satisfy what I will call here
(11) Mourrain’s condition. For f ∈ F , f ⊂ Π 1 (F ∩ Π<deg f ); i.e., in Mourrain’s words, F is connected to 1.
Mourrain’s condition implies that () 0 ∈ F but is, offhand, much stronger. For example, in the univariate case, (11) implies that F = Πk for some k, hence also that F is D-invariant, i.e., closed under differentiation. See [dB05b] for the fact that, in the multivariate case, (11) and D-invariance are not related. Mourrain [Mo] investigates the following problem: Given a finite-dimensional linear subspace F of Π and a linear projector N on Π 1 (F ) with range F , provide necessary and sufficient conditions on N to be the restriction to Π 1 (F ) of an ideal projector P with range F. There is at most one such ideal projector since, by Proposition 9, its kernel is necessarily the ideal generated by ker N. Mourrain shows the existence of such an ideal projector under the (obviously necessary) assumption that the linear maps
Mj : F → F : f 7 → N (()j f ), j = 1:d,
commute, but only for a F that satisfies (11).
and is a well-ordering, meaning that every subset of ZZd + has a smallest element. Standard examples are the Lexicographic Order (lex) in which α < β means that the first nonzero entry in β − α is positive, and the Graded Reverse Lexicographic Order (grevlex) in which α < β if, either |α| < |β|, or else |α| = |β| and the last nonzero entry in β − α is positive. Here and below,
|α| :=
j
α(j), α ∈ ZZd +.
Any such ordering admits the definition of the corresponding polynomial degree: Deg : Π\ 0 → ZZd + : p 7 → max supp ̂p,
with (13) ensuring that
(14) Deg(pq) = Deg(p) + Deg(q).
Note that, in this, the degree of the zero polynomial is undefined. Perhaps a mathematically cleaner definition of Deg(p) would be the set {α ∈ ZZd + : α ≤ max supp p̂} which now has the empty set as the natural definition of Deg(0) yet still satisfies (14) (since A + ∅ = ∅). With respect to such an ordering, one then constructs a Gr¨obner basis G for I, meaning that G is a finite subset of I with the property that
∀p ∈ I, p ∈
g∈G
g Π≤Deg(p)−Deg(g).
Here and below, for any subset Γ of ZZd + (including subsets merely specified by the condition its elements are to satisfy), ΠΓ := ran[()γ^ : γ ∈ Γ].
Actually, a simpler definition in use identifies a Gr¨obner basis for I as a finite subset G of I with ⋃
g∈G
(Deg(g) + ZZd +) ⊃ {Deg(f ) : f ∈ I} =: Deg(I).
Note that, directly from (14), Deg(I) = Deg(I) + ZZd +,
showing Deg(I) to be an upper set. But (by Dickson’s Lemma), any upper set U in ZZd + is necessarily of the form U = (∂U ) + ZZd +,
with ∂U := {α ∈ U : U \α is upper}
its necessarily finite boundary. This proves the existence of Gr¨obner bases. A naive definition of the normal form mod I for p ∈ Π is the element r of p + I of minimal Deg. However, there is, offhand, nothing to prevent I from containing f 6 = 0 with Deg(f ) < Deg(r), and then also (r + f )/2 is a different element of p + I of minimal degree. So, a better definition is the following. The normal form mod I for p ∈ Π is the unique element in (p + I) ∩ Π\ Deg(I).
Indeed, if both r and s are in this intersection, then their difference is in I, yet, if r − s were nonzero, then Deg(r − s) 6 ∈ Deg(I). This shows uniqueness. As to existence, let F := Π\ Deg(I) = ran[()α^ : α 6 ∈ Deg(I)].
Then, as we just pointed out, F and I are linear subspaces of Π with trivial intersection,
F ∩ I = { 0 }.
Further if, in the monomial order, the left shadow
ZZ≤α := {β ∈ ZZd + : β ≤ α}
of every α is finite (as is the case, e.g., in grevlex), then, for arbitrary p ∈ Π, the following elimination algorithm produces an r ∈ F with p − r ∈ I.
Division by G. Input: p ∈ Π, G. r ← p. for α = argmax(Deg(G) ∩ supp ̂r), and g ∈ G so that α = Deg(g), r ← r − (r̂(α)/̂ g(α))g. Output: The resulting r is “the remainder of the division of p by G”.
Indeed, for a monomial ordering such as grevlex, the entire calculation takes place on the finite index set ZZ≤Deg(p), hence necessarily stops after finitely many steps, at which point, assuming we chose G to be I, r ∈ F while, at every step, p − r ∈ I. For a monomial ordering, such as lex, in which left shadows can be infinite, a more subtle argument is required to prove that, nevertheless, the elimination algorithm terminates in finitely many steps. This more subtle argument leads naturally to the creation of a Gr¨obner basis G for I and its use in more refined versions of the elimination algorithm; see, e.g., [CLO92].
In any case, taking this for granted, we conclude that
Π = F ⊕ I,
with the normal form for p mod I nothing but the projection of p to F along I, i.e., the image of p under the ideal projector with range F and kernel I.
Note that F is quite a special algebraic complement for I. Not only is it monomial, in the sense that it is spanned by monomials, but, with that, F is also D-invariant, since Deg(I) = Deg(I) + ZZd +, hence
α 6 ∈ Deg(I) =⇒ (α − ZZd +) ∩ Deg(I) = ∅.
In other words, ZZd +\ Deg(I) is a lower set. This also implies that F satisfies Mourrain’s condition (11). Now, Mourrain’s point is that the construction of a Gr¨obner basis is, in general, time-consuming, as is working term by term. Can we, he asks, construct the normal form by some other, perhaps more efficient, way? If G spans an algebraic complement of some polynomial space F within Π 1 (F ), and if this F satisfies his condition (11) and is complementary to I = ideal(G), then, as we saw, for any p ∈ Π, its normal form mod I is the polynomial p(M )() 0 , with the Mj determined as above from the linear projector N on Π 1 (F ) with range F whose kernel is span(G). Mourrain also investigates the question of just what to do if we have to start with some arbitrary finite G, and develops an algorithm for constructing an H-basis for I = ideal(G), i.e., a finite subset H of I for which {h↑ : h ∈ H} is a basis for the homogeneous ideal
I↑ := ideal(p↑ : p ∈ I),
with p↑ uniquely determined (for p 6 = 0) by the requirements that it be homogeneous and satisfy
deg(p − p↑) < deg p,
and
deg p := max{|α| : p̂(α) 6 = 0}.
Lack of time and space prevents me from pursuing this further here. For H-bases in connection with multivariate polynomial interpolation, see [dB94], [MSa], [MSb], [MSc], [S98], [S01], [S02], [S05].
“Lefranc’s Nullstellensatz” [Le58]. For an arbitrary polynomial ideal I in Π = Π(Cd),
(16) I =
v
(I⊥v)⊥v ,
where, for any S ⊂ Π, S⊥v^ := {q ∈ Π : q(D)m(v) = 0, m ∈ S}
and S⊥v := {p ∈ Π : m(D)p(v) = 0, m ∈ S}.
Corollary. For an ideal projector P of finite codimension,
ran P ′^ =
v
δv Qv(D),
with Qv := I⊥v^ = {q ∈ Π : q(D)f (v) = 0, f ∈ I}.
Actually, the corollary can already be found in basic algebra books, e.g., [G70: p.168ff], but see already [G49] and the very nice overview article [G50]. Gr¨obner attributes the idea to Macaulay, e.g., [Mac: p.64ff], though it is described there in a different language (i.e., in terms of inverse systems) and there credit for first defining multiplicity correctly is given to Lasker [La05] (who, however, defines it only as a number, namely the length (i.e., the codimension) of the associated primary ideal). The space Qv = I⊥v^ is called the multiplicity space of I at v (or, less descriptively, the Max Noether space of I at v; see [MT]). Qv is a linear subspace of Π, of the same dimension as the linear subspace δv Qv(D) := {f 7 → q(D)f (v) : q ∈ Qv}
of Π′^ that it supplies, and, obviously,
δvQv(D) ⊂ I⊥^ = (ker P )⊥^ = ran P ′.
In other words, any ideal interpolant has interpolation conditions of the form
δvq(D)
for certain sites v and certain polynomials q. But much more is true. Since each of the spaces δvQv(D) lies in ran P ′, each must, in particular, be finite-dimensional. Also, since any finite sum of the form
∑
v
δv Qv(D)
is necessarily direct, there can be only finitely many nontrivial Qv here. But the most important fact is that each Qv is necessarily D-invariant. Is that obvious? It can be verified in many ways. Perhaps the simplest is the following which uses the intriguing formula
(17) q(D)f (0) =
α
Dαq(0)Dαf (0)/α! =: q ∗ f,
which, quite rightly, has made its appearance in various papers concerning multivariate polynomials but under various names (see, e.g., [S05: above Theorem 6.1]). It is the unique bilinear form on Π × Π for which
(18) (rq) ∗ f = q ∗ (r(D)f ), r, q, f ∈ Π.
(18) follows directly from (17) while, for the verification of (17), note that it is linear in q and f , hence can be verified by checking it for q = [[]]β^ : x 7 → xβ^ /β!,
the conveniently normalized power function, and f = [[]]γ^. For these, Dαq(0) = [[0]]β−α^ = δβ,α, hence
∑
α
Dαq(0)Dαf (0)/α! = δα,β δα,γ /α! = δβ,γ /β!,
while δ 0 ([[D]]β^ [[]]γ^ ) = δ 0 [[]]γ−β^ /β! = δγ,β /β!.
Note the symmetry, i.e., q ∗ f = f ∗ q,
hence, by symmetry, also (r(D)q) ∗ f = q ∗ (rf ).
Therefore, with Ev^ : f 7 → f (· + v)
the translation by v, we have, for q ∈ Qv, f ∈ I and r ∈ Π,
(r(D)q)(D)f (v) = r(D)q ∗ Evf = q ∗ (rEv^ f ) = q ∗ Ev((E−v^ r)f ) = q(D)((E−v^ r)f )(v) = 0,
since E−v^ r ∈ Π and therefore (E−v^ r)f ∈ I. With each Qv now known to be D-invariant, we know that it contains all constant polynomials if it is nontrivial. Hence, each nontrivial Qv supplies, in particular, the interpolation condition δv. The corresponding set V(I) := {v : Qv 6 = { 0 }}
is the variety of the ideal I, i.e., the set of zeros common to all polynomials in I. But, in general, we have not just the matching of function values, but also the matching of some derivative information, with the important restriction that, if δvq(D) is being matched, then so is δv(Dαq)(D) for all α. In the univariate case, there is only one D-invariant polynomial subspace of dimension k, namely Π<k, the polynomials of order k. But this says that, in the univariate case, ideal interpolation is Hermite interpolation. For that reason, we also use the term Hermite interpolation in the multivariate case when the interpolation conditions are of the form
∑
v
δv Qv(D)
with each Qv a D-invariant finite-dimensional polynomial space. Is any such Hermite interpolation ideal? If Q is any D-invariant linear subspace of Π, then, for arbitrary v, Q⊥v is an ideal: For, if q ∈ Q and f ∈ Q⊥v , then, for arbitrary r ∈ Π,
(rf ) ∗ Evq = f ∗ r(D)(Ev^ q) = f ∗ Ev^ (r(D)q) = 0,
since then r(D)q ∈ Q, hence also rf ∈ Q⊥v. But this says that
(
v
δvQv(D))⊥ =
v
(δv Qv(D))⊥ = ∩v Qv⊥v
is the intersection of ideals, hence an ideal. In other words, Hermite interpolation is characterized by the fact that it is ideal. Apparently, the first to use ‘Hermite interpolation’ in this sense in the multivariate context is H. M. M¨oller; see [M76], [M77] which predate [Bi79] and, in contrast to [Bi79], describe ran P ′. In [dBR90] and, regrettably, not yet aware of M¨oller’s work, we defined ‘Birkhoff-Hermite interpolation’ to mean that
(19) ran P ′^ = ∩v∈V δvQv(D),
Proposition 22 ([MSt]). The ideal projector P with F := ran P is Lagrange interpolation (i.e., #V = dim F ) if and only if the Mj are diagonalizable.
Proof: If #V = dim F , then, since dim ran P ′^ = dim F , [δv : v ∈ V] is an eigenbasis for M (^) p′ (for any p). Correspondingly, its dual basis in F , i.e., the basis [ℓv : v ∈ V] with
ℓv(w) = δvw, v, w ∈ V,
is an eigenbasis for Mp (again for any p); it is evidently the Lagrange basis for interpolation from F at V. Conversely, let V : Cn^ → ran P be an eigenbasis for the Mj. Then, the map
Π → Cn×n^ : p 7 → V −^1 p(M )V
is linear and, by (8), has ker P as its kernel. In other words, with λij the map that carries p ∈ Π to the (i, j)-entry of the matrix V −^1 p(M )V , we have
ker P = ∩i,j ker λij ,
hence (λij : i, j = 1:n) spans ran P ′. But, since V is an eigenbasis for the Mj , all the matrices V −^1 p(M )V are diagonal, hence only the λii are nontrivial and, since there are only n := dim ran P ′^ of them, they must form a basis for ran P ′. In particular, there must exist p ∈ Π for which #{λiip : i = 1:n} = n. Since {λiip : i = 1:n} = spect(p(M )) = {p(v) : v ∈ V}, this implies that #V = n.
As the simplest example, consider P : p 7 → p(0)()^0 + Dp(0)()^1. We compute the matrix representation for M 1 with respect to the standard basis, [()^0 , ()^1 ], for ran P = Π 1 (IF):
M 1 ()^0 = P ()^1 = ()^1 ; M 1 ()^1 = P (()^2 ) = 0,
hence
M̂ 1 = [ε 2 , 0] =
[
]
the simplest example of a defective matrix. It seems that Auzinger and Stetter [AS] were the first to propose to use the eigenstructure of the Mj for
the calculation of V. This requires, in principle, nothing more than the calculation of a matrix M̂ j similar to Mj , and this can be obtained in many ways, e.g., by computing the representation of Mj wrto some basis W of ran P. From this, one can, in principle, compute a basis U consisting of (generalized) eigenvectors for
any particular Mj , and, with that in hand, can now compute M̂ j := U −^1 Mj U for every j, hence know, in particular, not only v(j) for all j, but even the points v themselves, since one then knows the λii at least on Π 1. However, Auzinger and Stetter go for the eigenvectors of the transpose of M̂ j , as these are necessarily of the form δvU = (u(v) : u ∈ U ). Actually, [AS] focus on the left eigenvector av of the matrix M̂ p belonging to the eigenvalue p(v) since it is necessarily (a scalar multiple of) δvW , hence has w(v), w ∈ W , as its entries. If now W can be chosen to contain ()j , j = 1:d, then av contains the very coordinates of v. If W cannot be so chosen, still there are then techniques for teasing out v from the vector av; see [St], [MSt].
Is Hermite interpolation the limit of Lagrange interpolation?
While one is, of course, free to give names to hitherto unnamed concepts and constructs, use of an established name in a new or more general context needs justification. Since it is an integral and often used aspect of univariate Hermite interpolation that it is the (pointwise) limit of Lagrange interpolation, it is fair to ask whether multivariate ideal interpolation is also the limit of Lagrange interpolation. This question was already raised in [dBR90], within the restricted meaning of ‘Hermite interpolation’ used there, but has yet to be answered even in that restricted context.
To be sure, pointwise convergence of maps on a linear space depends on the notion of limit in that space to be employed. On Π, we use uniform convergence on compact sets or, what is the same, coefficient-wise convergence, i.e., lim n→∞ pn = p ⇐⇒ ∀α ∈ ZZd + lim n→∞
pn(α) = ̂p(α).
Proposition 23. The pointwise limit of ideal projectors is ideal.
Proof: Since the property of being ideal can be characterized pointwise (see Lemma 1), it is pre- served under pointwise convergence.
Since a linear projector is determined by its range and the range of its dual, the pointwise convergence of a sequence (Pn : n ∈ IN) of (finite-rank) linear projectors is equivalent to the convergence of their ranges and the ranges of their duals. Thus, we are interested in what limits, if any, can linear spaces spanned by finitely many point evaluations have as the evaluation sites all coalesce at one site, v. The above proposition implies that, if there is a limiting space, it is necessarily of the form δvQv(D) for some D-invariant space Qv. But the space Qv will crucially depend on just how the evaluation sites coalesce. Here is an example, from [dBR90].
Proposition 24. Let v and T be a point, respectively a finite subset, in ZZd. Then
lim h→ 0 ran[δv+hτ : τ ∈ T] = δvΠT(D),
with ΠT :=
p (^) T=
ker p↑(D).
Proof: Assume without loss that v = 0. Then the general element of ran[δv+hτ : τ ∈ T] is of the form λh : p 7 → λp(h·), with λ :=
τ ∈T
c(τ )δτ.
We compute
λhp = λp(h·) =
τ ∈T
c(τ )
α
(hτ )α^ p̂(α)
j
hj^
|α|=j
τ ∈T
c(τ )τ α ︸ ︷︷ ︸ λ()α
p̂(α)
j≥ord λ
hj^
|α|=j
λ()α^ p̂(α)
with ord λ := min{|α| : λ()α^6 = 0}.
Therefore
lim h→ 0 λhp/hord^ λ^ =
|α|=ord λ
λ()α^ p̂(α) =
|α|=ord λ
λ()α^
α!
Dαp(0)
= q(D)p(0),
with
q :=
|α|=ord λ
τ ∈T
c(τ ) τ α α!
()α^ = ???
a certain polynomial. Note that, in the univariate case, this sum would only have one term in it and, correspondingly, the limit is just a scalar multiple of the (ord λ)-th derivative at the origin, just as expected. In the multivariate case, things are more complicated. Yet, as we look further into this polynomial q, we’ll also discover real beauty.
The equivalence of Π′^ with P claimed in (25) can be established in several ways. For our purposes, it is convenient to do it via the natural extension of the bilinear form (17) to
P × Π → IF : (f, p) 7 → f ∗ p =
α
f̂ (α)α!p̂(α).
Note that, for any v ∈ IFd^ and any p ∈ Π,
ev ∗ p =
α
vα^ p̂(α) = p(v).
In other words, the exponential function with frequency v represents evaluation at v with respect to this pairing. In particular, given that we were interested in finding limh→ 0
τ c(τ^ )δhτ^ , the appearance of the exponential function in the above proof is not accidental. Note further that ΠT is not only D-invariant (as the intersection of kernels of constant-coefficient differential operators) but also dilation-invariant (as the span of homogeneous polynomials). In contrast, in general, the multiplicity spaces Qv for an ideal projector need only be D-invariant. Here is a further example, from [dBR90], to show how such a δvQv(D) may, nevertheless, be the limit of spaces spanned by point evaluations. Let Th := {ξ− := (−h, h^2 ), 0 , ξ+ := (h, h^2 )} ⊂ IF^2 and set Mh := ran[δτ : τ ∈ Th]. Then, with ξ 0 := (0, h^2 ), Mh contains
(δξ+ + δξ− − 2 δ 0 )/h^2 = (δξ+ − 2 δξ 0 + δξ− )/h^2 + 2(δξ 0 − δ 0 )/h^2 ,
and this evidently converges to δ 0 (D^21 + 2D 2 ) as h → 0, while certainly (δξ+ − δξ− )/h is in Mh and converges to δ 0 D 1 , and δ 0 is in Mh for all h. This shows that the 3-dimensional space δ 0 Q 0 (D) with Q 0 := ran[()^0 , ()^1 ,^0 , ()^2 ,^0 + 2()^0 ,^1 ] is in limh→ 0 Mh, hence must coincide with it since each Mh is only 3- dimensional. Note that Q 0 is D-invariant but not dilation-invariant.
Conjecture. A linear projector on Π ⊂ (Cd^ → C) is ideal if and only if it is the (pointwise) limit of Lagrange interpolation.
Some people have told me that this conjecture is obviously true, because of known results concerning the resolution of singularities. On the other hand, Geir Ellingsrud has pointed out to me that this conjecture must fail for d > 2, because of results by Iarrobino (see [I]) concerning the dimension of the manifold of ideals of codimension k with k points in their variety as compared with the dimension of the manifold of ideals of codimension k with variety { 0 }. But, lacking as yet a sufficiently good background in Algebraic Geometry, I have not yet understood his reasoning. In any case, Ellingsrud’s remark does not contradict the following, very recent, response, by Boris Shektman, to the above conjecture.
Proposition 26 ([Sh]). Any ideal projector on Π ⊂ (C^2 → C) with range the polynomials of degree ≤ k (for some k) is the pointwise limit of Lagrange interpolation projectors.
Proof outline: Let F = Πk be the range of the ideal projector P , and recall from Proposition 22 that P is Lagrange interpolation iff the linear maps Mj : F → F : f 7 → P (()j f ) are diagonalizable. Since F is finite-dimensional, the diagonalizable linear maps on F are dense in L(F ). Hence we are looking for an indication that the set of all ideal projectors with range F is open in some sense. From Proposition 7, we know that P is characterized by its action on Π 1 (F ) = Πk+1, hence by the polynomials hα := P ()α^ ⊂ ran P = Πk, |α| = k + 1,
since P ()α^ = ()α^ for |α| ≤ k. On the other hand, while any choice of the hα gives rise to a linear projector N on Π 1 (F ) with range F = Πk, not all of them are the restriction to Π 1 (F ) of an ideal projector with range F. Since F evidently satisfies Mourrain’s condition (11), we know from Theorem 12 that N is the restriction of an ideal projector with range F if and only if
N (()iN (()j ()α)) = N (()j N (()i()α)), |α| ≤ k, 1 ≤ i < j ≤ d.
Now, for |α| < k, N (()i()α) = ()i()α, hence the condition is equivalent to
N (()iN (()j ()α)) = N (()j N (()i()α)), |α| = k, 1 ≤ i < j ≤ d.
Further, for |α| = k, N (()i()α) = hεi +α, hence the condition is that
()ihεj +α − ()j hεi+α ∈ ker N, |α| = k, i < j.
But (()β^ − hβ : |β| = k + 1) is evidently linearly independent (since hβ ∈ Πk) and has dim ker N terms and is in ker N , hence is a basis for ker N. Therefore, the choice (hβ : |β| = k + 1) specifies an ideal projector with range Πk if and only there are matrices Cij (necessarily unique) so that
(27) ()ihεj +α − ()j hεi +α =
|β|=k+
Cij (α, β)(()β^ − hβ ), |α| = k, i < j.
Now, in the bivariate case actually under discussion, there is just one choice for (i, j), namely (1, 2), hence (hβ : |β| = k + 1) in Πk gives rise to an ideal projector with range Πk if and only if there is some matrix C so that
(28) () 1 hε 2 +α − () 2 hε 1 +α =
|β|=k+
C(α, β)(()β^ − hβ ), |α| = k.
It is this equation, Shekhtman derives and looks at. He treats it as an equation for the vector h := (hβ : |β| = k + 1), hence writes it in the form
Ah − C(b − h) = 0,
with b := (()β^ : |β| = k + 1)
and Ah := (() 1 hε 2 +α − () 2 hε 1 +α : |α| = k),
hence Ab = 0,
therefore (28) is equivalent to
(29) (A + C)(h − b) = 0.
Now, given that A + C has one more column than it has rows, it follows, by a standard formula, that
(30) h := (()β^ − (−1)β^ det(A + C)(:, \β) : |β| = k + 1)
solves (29), hence (28). Further, det A(:, \β) = (−1)β^ ()β^ , hence this h is in Πk, as required. This shows that each choice of C gives rise to an ideal projector. It also shows that each det A(:, \β) is nonzero almost everywhere, hence A + C is onto almost everywhere and, therefore, ker(A + C) is 1-dimensional almost everywhere. In other words, for given C, h uniquely solves (28). Now notice that (30) describes the solution h as a polynomial function in the entries of the matrix C. Hence, with Λ a basis for ran P ′^ and n := dim F = dim ran Λ, the determinant of the Gram matrix
Λt[()j 1 : j < n]
is also a polynomial in the entries of C, and is nonzero for some choice of C. Hence, every neighborhood of our ideal projector P contains an ideal projector R with range Πk and such that, for any basis M for ran R′, the Gram matrix Mt[()j 1 : j < n] is invertible, hence there is a linear projector S with ran S = ran[()j 1 : j < n] and ran S′^ = ran R′, hence an ideal projector. By perturbing, if necessary, the zeros of the polynomial ()n 1 − S()n 1 (considered as a univariate polynomial), we obtain (see the example following Proposition 7) an interpolating ideal projector T as close to S as we would like, and, with that, the linear projector U with range Πk and ran U ′^ = ran S′^ is well-defined and an interpolating projector as close to P as we would like.
with b := (· − τ 1 ) · · · (· − τn)
the monic polynomial that vanishes at the interpolation sites to the appropriate multiplicity, i.e., the monic polynomial that generates the ideal ker P , and ∆(τ 1 ,... , τn, x)f the divided difference of f at the sites τ 1 ,... , τn, x, hence a polynomial in x that depends linearly on Dnf. More precisely,
∆(τ 1 ,... , τn, x)f =
K(·|τ 1 ,... , τn, x)Dnf
for a certain function K, namely a B-spline with knots τ 1 ,... , τn, x. Since Dn^ = b↑(D), one may therefore hope, in the multivariate case, for an error formula of the form
(31) f (x) − P f (x) =
b∈B
b(x)Ix,b(b↑(D)f )
with B a minimal generating set for I and with each Ix,b some linear integral operator. Since ran P comprises exactly those polynomials for which f − P f = 0, this would imply ⋂
b∈ker P
ker b↑(D) =
b∈B
ker b↑(D) ⊆ ran P,
the equality holding because B is a basis for the ideal ker P. But since ran P is complementary to the ideal ker P , this would imply (^) ⋂
p∈ker P
ker p↑(D) = ran P.
But this implies (see [dBR92a]) that P is necessarily the least projector for the given interpolation conditions (ker P )⊥, as introduced in [dBR92a] for arbitrary (finite-dimensional) spaces of interpolation conditions. I resist the urge to call the linear projector with
ran PI =
p∈I
ker p(D) and ker PI = I
a ‘least ideal projector’, and call it least Hermite interpolation instead. As a simple example, consider interpolation at Σ × T, with Σ and T finite subsets of IF. The ideal I of all bivariate polynomials vanishing on Σ × T is generated by the two polynomials
pσ : x 7 →
σ∈Σ
(x(1) − σ), pτ : x 7 →
τ ∈T
(x(2) − τ ).
Correspondingly, with m := deg pσ , n := deg pτ ,
the least choice for the space from which to interpolate in this case is the standard one, i.e.,
ran PI = ker(pσ )↑(D) ∩ ker(pτ )↑(D) = ker D 1 m ∩ ker Dn 2 = ran[()α^ : α(1) < deg pσ, α(2) < deg qτ ].
However, the standard formula for the error in such tensor-product interpolation to f involves not only Dm 1 f and Dn 2 f but also the higher mixed derivative Dm,nf. Nevertheless, it is possible (see [dB97]) to derive an error formula for this particular, and even for general multivariate, tensor product interpolation, of the form (31), with B the ‘natural’ basis for I. But (31) fails the next test, Chung-Yao interpolation, for which the error formula, derived in [dB97], is of the slightly more complicated form
(32) f (x) − P f (x) =
b∈B
b(x)Ib,x(˜b↑(D)f ),
with (˜b : b ∈ B) also a (minimal) basis for I and such that ˜b↑(D)c = δb,c for b, c ∈ B. One may therefore hope for an error formula of the form (32) for arbitrary least Hermite interpolation (a hope first expressed in [dB97]). But, already for general Lagrange interpolation from Πk, this is still only a hope, as the Sauer-Xu error formula for that case (see [SX95a]) does not readily convert into the form (32).
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