A Generalization of the Remainder in Multivariate Polynomial

Interpolation

Dana Simian

Department of InformaticsFaculty of Sciences

The ”Lucian Blaga” University of SibiuSibiu 5-7 dr. I.Ratiu str.

ROMANIA

Abstract: The aim of this article is to introduce a generalization of the interpolation remainder in multivariateinterpolation and to study it, in least interpolation schemes and in minimal interpolation ones. This generalization,we named λ-remainder, allows us a deeper analysis of the error in the interpolation process. Connected to theλ - remainder, we introduced the notion of λ - error order of interpolation. In the end we provide particularapplications of the results we obtained. Interesting additionally theorems are also proved.

Keywords: multivariate polynomial interpolation, λ - remainder, λ - error order of interpolation

1 Introduction

Interpolation by polynomials in several variables isan active area of research. A survey of the main re-sults on multivariate polynomial interpolation in thelast thirty years can be found in [?]. An importantproblem in the field of interpolation is the construc-tion of a polynomial interpolation space for arbitraryinterpolation nodes and the estimation of the interpo-lation remainder. In 1990, de Boor and Ron, in [?],constructed, for a given set of arbitrary nodes, an in-teresting polynomial interpolation space, named ”leastinterpolation space”. This construction admits a ge-neralization for an arbitrary set of conditions. We re-call now, the general formulation of the interpolationproblem, we present the construction of the ”least in-terpolation space” for an arbitrary set of functionals(introduced in [?]) and then we prove a theorem weneed in the next section.

Let Λ = {λ1, . . . , λn} be a set of linear functionals,linear independent, Πd the space of polynomials in dvariables and F a space of functions which includespolynomials. The polynomial interpolation problemwith respect to the set of conditions Λ is to constructa polynomial subspace P such that for an arbitrary

function f ∈ F there exists a unique polynomial p ∈ Psatisfying the conditions λ(p) = λ(f), ∀λ ∈ Λ. In thiscase, we say that the pair (Λ,P) is correct, or, equiva-lently, P is an interpolation space with respect to Λ.

We can associate to any functional λ, its formalpower series or its generating function,

λν =∑

α∈Nd

λ(mα)α!

xα, (1)

with, mα(x) = xα, x ∈ Rd, α ∈ Nd.Obviously, for any p ∈ Πd, one has

λ(p) = (p(D)λν)(0) (2)

We define, for any power series f (or analyticalfunction), its least term, f ↓, which is its nonzero ho-mogeneous term of minimal degree.

”Least interpolation space”, with respect to the setof functionals Λ, is the polynomial subspace generatedfrom the least terms of the generating functions of thefunctionals in Λ:

HΛ↓= span{λν↓ | λ ∈ Λ} (3)

It is proved in [?], that HΛ ↓ is a minimal interpola-tion space, or equivalent a degree reducing interpola-

tion space.A particular choice of functionals leads us to many

interesting interpolation spaces.In order to get an expression of the λ-remainder

in least interpolation we need some additional results.First we introduce the pair between an analytical func-tion and a polynomial:

< f, p >= (p(D)f)(0) (4)

which is a veritable inner product on polynomial spa-ces.

Proposition 1 (de Boor, [?]) There always is a basisgi, i ∈ {1, . . . , #HΛ} of the space

HΛ = span{λν | λ ∈ Λ},which is orthogonal to HΛ↓ in the sense of pair (??),that is < gi, gk↓>6= 0 ⇔ k = i.

The basis gi can be constructed using aGramm - Schmidt type algorithm, (see [?]), or a Gausselimination by segments algorithm. The description ofthe idea of the Gauss elimination by segments algo-rithm can be found in [?]. Using this basis, we canextend the product from (??) in the following sense:let cj,i be the coordinates of gj in the basis λν of HΛ,that is

gj(x) =n∑

i=1

cj,i λνi (x), (5)

then

< gj , f >=n∑

i=1

cj,iλi(f), ∀f ∈ A0 (6)

A generalization of the results from [?] is given inthe following theorem

Theorem 1 The unique element LΛ(f) ∈ HΛ↓ whichinterpolates f ∈ A0 with respect to the conditions Λ is

LΛ(f) =n∑

j=1

gj↓ < gj , f >

< gj , gj↓>, n = #Λ (7)

Proof: Taking into account that

< gi, LΛ(f) >=< gi, f >, ∀ f ∈ A0,

we get λk(LΛ(f)) = λk(f), k ∈ {1, . . . , n}. The unicityof LΛ follows from the interpolation property of thespace HΛ↓. ♠Theorem 2 ([?], [?]) The operator

L∗Λ(f) =n∑

j=1

gj< f, gj↓>< gj , gj↓> ; f ∈ A0. (8)

has the duality property:

< L∗Λ(g), f >=< g, LΛ(f) >, g, f ∈ A0, (9)

Theorem 3 The operator LΛ satisfies the inequality:degLΛ(f) ≤ deg(f), ∀ f ∈ Πd and the inequality isstrict if and only if f↑ ⊥HΛ↓, with f↑ being the leadingterm of the polynomial f .

Proof: There comes out from (??), thatdeg(LΛ) ≤ max(deg(gj ↓)). If deg(gj ↓) > k then< gj , f >= 0. If deg(gj ↓) = k, then, the followingimplications hold:f↑⊥ HΛ↓⇔< p, f↑>= 0,∀ p ∈ HΛ↓⇒< gj↓, f↑>= 0 ⇒< gj , f >= 0, ∀ j ∈ {1, . . . , n}

Consequently deg(LΛ(f)) < deg(f), ∀ f ∈ Πd,with f↑⊥ HΛ↓.

Let’s suppose now that deg LΛ(f) < deg (f).Hence (f − LΛ(f))↑= f↑.

We use the fact that if p(D) annihilates HΛ, thenp↑ (D) annihilates HΛ↓ (see [?]) and the following im-plications:

λ(p) = 0, ∀ λ ∈ Λ ⇔ p ⊥ HΛ ⇒ p↑⊥ HΛ↓ .

Therefore:λ(f − LΛ(f)) = 0, ∀ λ ∈ Λ ⇔ (f − LΛ(f)) ⊥ HΛ

⇒ (f − LΛ(f))↑⊥ HΛ↓⇔ f↑⊥ HΛ↓ .♠

2 The λ -remainder and λ-error order of in-terpolation

Let’s consider the general polynomial interpolationproblem with conditions Λ and let LΛ be the correspon-ding interpolation operator and RΛ be the remainderoperator. The interpolation formula is:

f = LΛ(f) + RΛ(f) (10)

Definition 1 We name λ-remainder, the value

RΛ,λ(f) = λ[(1− LΛ)(f)]; f ∈ A0; λ ∈ Π′ (11)

Consequently, for any x ∈ Rd, the classical remain-der (RΛ(f))(x) is in fact the δx-remainder. For anyfunctional λ ∈ Λ we obtain RΛ,λ(f) = 0, ∀f ∈ A0.

Definition 2 (C. de Boor, [?]) Let be L : A0 → Πd apolynomial interpolation operator. The error order ofinterpolation is the greatest integer k such thatf(x)− (L(f))(x) = 0, ∀ f ∈ Πd

<k.

We generalize this definition.

Definition 3 We name λ-error order of interpolationthe greatest integer k such that

RΛ,λ = 0, ∀ f ∈ Πd<k, λ ∈ (Πd)′,

with RΛ,λ the λ - remainder, defined in (??).

If in definition ?? we take λ = δx, ∀x ∈ Rd, weobtain definition ??.

2.1 Case of least interpolation scheme

Proposition 2 The λ-remainder in ”least interpola-tion” scheme can be expressed such as:

RΛ,λ(f) =< ενΛ,λ, f >, (12)

withενΛ,λ = (1− L∗Λ)(λν). (13)

Proof:RΛ,λ(f) = λ[(1− LΛ)(f)] =< λν , (1− LΛ)f >==< λν , f > − < L∗Λ(λν), f >=< εν

Λ,λ, f >. ♠Corollary 1 ([?]) The expression of the classical in-terpolation remainder is:

(R(f))(x) =< ex − L∗Λ(ex), f >, (14)

with ex(t) = ext; x, t ∈ Rd.

Theorem 4 ενΛ,λ ⊥ HΛ↓ and every homogeneous com-

ponent of ενΛ,λ satisfies the same orthogonality proper-

ty.

Proof: Let be p ∈ HΛ↓. Then,

< ενΛ,λ, p >= RΛ,λ(p) = 0.

Consequently, ενΛ,λ ⊥ HΛ↓.

The polynomial subspace HΛ ↓ is generated byhomogeneous polynomials. Therefore (εν

Λ,λ)[k] ⊥ HΛ↓.For any analytical function, g ∈ A0, we had definedthe k - order homogeneous component, such as:

g[k] =∑

|α|=k

Dαg(0)(·)α/α!

Proposition 3 ενΛ,λ satisfies the equality

< ενΛ,λ, f >= λ(f),∀ f ∈ kerLΛ (15)

Proof: < ενΛ,λ, f >=< λ, f > − < λ, LΛ(f) >.

f ∈ kerLΛ ⇒ LΛ(f) = 0 ⇒< λ, LΛ(f) >= 0. ♠The following theorem gets the λ-error order of in-

terpolation in the particular case of least interpolationscheme.

Theorem 5 If HΛ↓6= Πm, then the λ-error order ofinterpolation is given by the deg(εν

Λ,λ↓), ενΛ,λ being de-

fined in (??).

Proof: Let k = deg(ενΛ,λ↓) and f ∈ Πd

<k. Obviouslydeg(f ↑) < k and < εν

Λ,λ, f >= 0, hence the λ-errororder of interpolation is greater than or equal to k.

First, let’s consider that there is not any m ∈ Nsuch that HΛ↓= Πd

m. Let’s suppose by contradictionthat RΛ,λ(f) = 0,∀ f ∈ Πd

k. Then < ενΛ,λ, f >= 0 and

taking into account theorem ?? we also get

< ενΛ,λ↓, f >= 0,∀ f ∈ Πd

k.

This is a contradiction, because ενΛ,λ↓ is a homogeneous

polynomial of degree k.Similarly, the supposition that RΛ,λ(f) = 0,

∀ f ∈ Πd<q, with q > k leads us to the contradiction

< ενΛ,λ↓, f [k] >= 0, ∀ f ∈ Πd

<q.If HΛ↓= Πd

m, the λ- error order of interpolation ism + 1, be cause εν

Λ,λ↓⊥ HΛ↓. ♠Theorem 6 Let be

P = {p ∈ Πd| λ(p) 6= 0, λ 6∈ Λ; p ∈ ker(LΛ)}Then the following equality holds:

deg ενΛ,λ↓= min{deg p|p ∈ P}.

Proof: Let denote by

k = deg ενΛ,λ↓ and k′ = min{deg p|p ∈ P}

Taking into account theorem ?? we get:p ∈ ker(LΛ) ⇒< εν

Λ,λ, p >= λ(p) 6= 0⇒ deg εν

Λ,λ↓≤ deg p↑= deg p ⇒ k ≤ k′.On the other hand let q = εν

Λ,λ↓ −LΛ(ενΛ,λ↓). Using

theorems ?? and ??, we obtain

deg q = deg ενΛ,λ↓= k.

Much more λ(q) = RΛ,λ(ενΛ,λ↓) =< εν

Λ,λ, ενΛ,λ↓> > 0

and LΛ(q) = 0. But, from q ∈ Πd; λ(q) 6= 0 andq ∈ ker(LΛ) we obtain deg q ≥ k′, that is k ≥ k′. ♠Corollary 2 If q ∈ Πd

≥k, then the expression of the λ-remainder is

RΛ,λ(q) =∑

α∈Nd,|α|≥k

DαενΛ,λ(0) ·Dαq(0)

α!

with k = deg ενΛ,λ↓.

Corollary 3 ενΛ,λ vanishes to order k = deg εν

Λ,λ↓ at0.

Proof: It easily results from the equality(q(D)εν

Λ,λ

)(0) = 0,∀ q ∈ Π<k

The next theorem allows us to study the classicalremainder and hence to obtain the error order of inter-polation in least interpolation.

Theorem 7 The operator LΛ given in (??) reproducesthe monomials xα, x ∈ Rd, α ∈ Nd, if and only ifn∑

i=1

ck,iλi(xα) = Dαgk(0), with coefficients ck,i given

in (??).

Proof:n∑

i=1

< gk, ϕiλi(xα) >=n∑

i=1

λi(xα)cki.

On the other hand,n∑

i=1

< gk, ϕiλi(xα) >=< gk, xα >= D(α)gk(0)

2.2 Case of minimal interpolationschemes

A finite set, Λ, of linear functionals is said to admitan ideal interpolation scheme if ker(Λ) is a polynomialideal.

Definition 4 ([?],[?]) A polynomial subspaceV ⊂ Πd

n is a minimal interpolation space of order nwith respect to Λ if it is a degree reducing interpola-tion space and the set of conditions Λ admits an idealinterpolation scheme.

As pointed out by many authors (de Boor in [?], Sauerin [?]), ideal interpolation schemes can even be charac-terized as Hermite interpolation schemes with an ad-ditional closedness condition. Minimal interpolationspaces are deeply connected with the notion of New-ton basis.

Definition 5 We say that the polynomial spaceV ⊂ Πd

n admits a Newton basis of order n with res-pect to the set of functionals Λ, if the functionals inΛ may be reindexed in the blocks Λ(k) = {λα : λα ∈Λ; α ∈ Ik \ Ik−1}, using a grading set of multiindicesI = (I0, . . . , In), Ik \ Ik−1 ⊂ {α : |α| = k};k = 0, . . . , n, such that1. There is a basis pα ∈ Πd

|α|, α ∈ In of P(Λ) with

λβ(pα) = δα,β ; β ∈ In; |β| ≤ |α| (16)

2.There are the complementary polynomials

p⊥α ∈ Πd|α| ∩ker(Λ), α ∈ I ′n = {α ∈ Nd : |α| ≤ n} \ In,

satisfying

Πdn = span{pα : α ∈ In} ⊕ span{p⊥α : α ∈ I ′n}

The number of functionals in the block Λ(k) equalsthe dimension of homogeneous subspace of V , of degreek.

It is known ( see [?]) that a polynomial subspace isa minimal interpolation space of order n with respectto Λ if and only if it admits a Newton basis of order nwith respect to Λ.

In order to compute the λ-remainder in minimalinterpolation schemes, we need to construct the corres-ponding Newton basis of the interpolation space. Wecan do this, in an inductive way, using a generalizationof the method presented in [?], if we know a basis of theminimal interpolation space. We will denote the func-tionals in block Λ(k), by λ

[k]i , 1 ≤ i ≤ nk, Vk = V ∩Πd

k

and Qk = Q ∩Πdk.

V is a degree reducing interpolation space, henceΠd

n = V ⊕ Qn and Πdk = Vk ⊕ Qk. Therefore, there is

a grading basis of V , g1, . . . , gN , with N = dim V .

Consequently, we may define a system of set of mul-tiindices I = (I0, . . . , In) so that I0 ⊂ I1 ⊂ . . . ⊂ In

and #Jk = nk = dim V 0k , with V 0

k being the homoge-neous subspace of V , of k order, Jk = Ik \ Ik−1 and wemay rewrite the grading basis like {gα : α ∈ Jk; k =0, . . . , n}.

We find the blocks Λ(k), reindex the functionals inthese blocks and starting from them, we construct thespaces Vk and Qk.

We choose the right functionals λ[s]r ;

s ∈ {0, . . . , n}; r ∈ {1, . . . , ns} and construct, for everypair (j, k), k ∈ {0, . . . , n}, j ∈ {1, . . . , nk}, the poly-nomials p

[l]i ∈ Πd

l ; l ∈ {0, . . . , n}; i ∈ {1, . . . , nl} suchthat, for (r, s) ≤ (i, l) ≤ (j, k):

λ[s]r (p[l]

i ) = δl,s · δi,r, (17)

and the polynomials q[l]i , with i ∈ {j, . . . , rl}, such that

λ[s]r (q[l]

i ) = 0, for (r, s) ≤ (j, k) < (i, l) (18)

We will use the double induction, first on k and,for a certain k, induction on j: 0 ≤ k ≤ n; 1 ≤ j ≤ nk.

We initialize q[l]i = g

[l]αi ; l = 0, . . . , n; i ∈ {1, . . . , nl}

and complete the set of polynomials q[l]i for i =

nl + 1, . . . , rl = dim Πdl to a basis for Πd

n. The cor-rectness of the pair (Λ, V ) implies the existence of afunctional λ

[0]1 ∈ Λ so that λ

[0]1 (g

α[0]1

) 6= 0. Then,

p[0]1 =

1

λ[0]1 (g

α[0]1

)and q

[l]i = g

α[l]1− λ

[0]1 (g

α[0]1

);

(1, 0) < (i, l); i = 1, . . . rl

Let us suppose that for certain k and j, 0 < k < n,1 < j < nk we have already done the required con-struction. Let Λ = Λ \ {λ[l]

i : (i, l) ≤ (j, k)} the set offunctionals that have not yet put into blocks. Again,the correctness of pair (Λ, V ), implies the existence ofa functional λ

[k]j+1 ∈ Λ so that λ

[k]j+1(q

[k]j+1) 6= 0 ( if not

q[k]j+1 vanishes on all Λ).

We set

p[k]j+1 =

q[k]j+1

λ[k]j+1(q

[k]j+1)

which satisfies λ[k]r (p[k]

j+1) = δj+1,r, ∀r ≤ j + 1. Thepolynomials

p[k]i = p

[k]i − λ

[k]j+1(p

[k]i ) · p[k]

j+1; i = 1, . . . , j

satisfy λ[k]j+1(p

[k]i ) = 0 and λ

[k]r (p[k]

i ) = 0, ∀r < j, hence,

replacing p[k]i , i ∈ {1, . . . , j} with p

[k]i we obtain polyno-

mials and functionals which satisfy (??) with j → j+1.The polynomials

q[l]i = q

[l]i − λ

[k]j+1(q

[l]i ) · p[k]

j+1, (j + 1, k) < (i, l)

satisfy

λ[s]r (q[l]

i ) = 0, for (r, s) ≤ (j + 1, k) < (i, l)

Hence, replacing the polynomials q[l]i with q

[l]i for

(j + 1, k) < (i, l), we obtain polynomials which satisfy(??) for j → j + 1.

This finish induction on j. Similarly, it may bedone the induction on k.

Definition 6 Let (pα), α ∈ In be the Newton basisfor the minimal interpolation space of n order V andΛ(k) the proper blocks of functionals. The λ-divideddifference is defined recursively by:d0[λ; f ] = λ(f)dk+1[Λ(0), . . . , Λ(k), λ; f ] =

= dk[Λ(0), . . . , Λ(k−1), λ; f ]−−

∑

α∈Jk

dk[Λ(0), . . . , Λ(k−1), λα; f ]λ(pα)

with Jk = Ik \ Ik−1.

Taking Λ = {δθ : θ ∈ Θ}, where δx is the eva-luation functional in x ∈ Rd, we obtain the divideddifference used by T. Sauer in [?].

With the notations in the definition ??, we get:

Theorem 8 The λ-remainder in interpolation from aminimal interpolation space of order n has the expres-sion:

RΛ,λ(f) = dn+1[Λ(0), . . . , Λ(n), λ; f ] (19)

The proof may be done using induction on n.

3 Application

Let Λ = δΘ,

Θ = {θi; i = 1, . . . , 4} = {(a, 0); (0, b); (−a, 0); (0,−b)}a, b ∈ R+. We generate the basis {gi; i = 1, . . . , 4} and{gi↓; i = 1, . . . , 4} of the spaces HΛ and HΛ↓, using aGramm-Schmidt type algorithm or Gauss eliminationby segments (see [?],[?]). We start with g1 = eθ1 andobtain:g1(x, y) =

1a4 + b4

[b4 cosh(ax) + a4 cosh(by)

]

g1↓ (x, y) = 1; < g1, g1↓>= 1g2(x, y) = sinh(by)− sinh(ax)g2↓ (x, y) = by − ax; < g2, g2↓>= a2 + b2

g3(x, y) =1

(a2 + b2)(a4 + b4)· E(x, y)

E(x, y) = [(a6 − a4 + a4b2 + a2b2 + 2b6) cosh(ax)−−2(a4b2 + b6)eax++(a6 + a4 + a4b2 − a2b2 + 2a2b4) cosh(by)−−2(a6 + a2b4)eby]

g3↓ (x, y) = − 2ab2

a2 + b2x− 2a2b

a2 + b2y;

< g3, g3↓>=4a2b2

a2 + b2

g4(x, y) = 2[cosh(by)− cosh(ax)]g4↓ (x, y) = −a2x2 + b2y2;< g4, g4↓>= 2(a4 + b4).

The polynomials g2↓ and g3↓ are linear indepen-dent, hence, the interpolation space is:HΛ↓= Π1 + span{−a2x2 + b2y2}

The coefficients ci,j ; i, j ∈ {1 . . . 4}, in formula (??)are given below:

c1,1 =b4

2(a4 + b4); c2,1 = −1/2;

c1,2 =a4

2(a4 + b4); c2,2 = 1/2;

c1,3 =b4

2(a4 + b4); c2,3 = 1/2;

c1,4 =a4

2(a4 + b4); c2,4 = −1/2;

c3,1 = k(a6 − 3a4b2 − a4 + a2b2 − 2b6);c3,2 = k(−3a6 + a4b2 + a4 − a2b2 − 2a2b4);c3,3 = k(a6 + a4b2 − a4 + a2b2 + 2b6);c3,4 = k(a6 + a4b2 + a4 − a2b2 + 2a2b4);

c4,1 = −1; c4,2 = 1;c4,3 = −1; c4,4 = 1;

We used the notation k =1

2(a2 + b2)(a4 + b4).

We will calculate and analyze the λ-error order ofinterpolation for various choices of functional λ.Case 1. The case of evaluation functionals

We want to study the classical error order of inter-polation, using theorem ??. In this case λ = δ(t1,t2),(t1, t2) ∈ R2. The generating function is λν(x, y) =et1x+t2y. We calculate the homogeneous componentsof εν

Λ,λ and obtain:(εν

Λ,λ)[0] = 0;

(ενΛ,λ)[1] = b(b−a)

a2+b2 (t1 − t2)x++ 1

a2+b2 [t1(ab− a2)− t2(ab + b2)]yIf a 6= b then deg(εν

Λ,λ)↓= 1, ∀(t1, t2) ∈ R2 and theλ-error order equals 1, ∀λ = δ(t1,t2), that is the classi-cal error order of interpolation is equal to 1.

If a = b, then for t2 = 0, (ενΛ,λ)[1] = 0, that is

the λ-error order of interpolation is greater than 1, forλ = δ(t1,0).

The same result can be obtained using theorem ??.LΛ reproduces the constant functions, because

4∑i=1

ck,iδθi(1) = gk(0), ∀k ∈ {1, . . . , 4}, but it does not

reproduce the polynomials of degree 1, because

4∑

i=1

c3,iδθi(m(1,0)) 6= D(1,0)g3(0).

Consequently, the error order of interpolation is equalto 1.

We notice that the analysis of λ- error order ofinterpolation is deeper than the analysis of classicalerror order of interpolation.Case 2. The case of Birchoff type functionals

The functional λq,θ(f) = (q(D)f)(θ), q ∈ Πd, θ ∈R2 is a generalization of the derivative of f at the pointθ, that is why, it is important to study the λq,θ - errororder of interpolation.

Let’s choose q(x, y) = m(1,0)(x, y) = x and θ =(t1, t2) ∈ R2. The generating function is λν

q,θ(x, y) =xet1x+t2y.

Next, we will denote λ = λm(1,0),(t1,t2). The resultswe have obtained are:

(ενΛ,λ)[0] = 0; (εν

Λ,λ)[1] = 0;

(ενΛ,λ)[2] = x2

[b4

a4+b4 t1 + +a(a6−a4−a4b2+a2b2)4(a2+b2)(a4+b4)

]+

+t2xy + y2[

a2b2

a4+b4 t1 + b2(−a6+a4+a4b2−a2b2)4a(a2+b2)(a4+b4)

]

That means that deg(ενΛ,λ) = 2 and the λ-error or-

der of interpolation is equal 2. If t2 = 0 and a6−a4b2−b6 − b4 = 0 we have deg(εν

Λ,λ) > 2 and in this case theλ-error order is greater then 2.

For the set of functionals Λ, we considered in thissection, obviously ker(Λ) is a polynomial ideal, that isHΛ↓ is a minimal interpolation space of order 2. Hence,we can apply the results obtained in subsection ??. Wewill have three blocks of functionals:Λ(0) = {λ[0]

1 }; λ[0]1 = λ(0,0)

Λ(1) = {λ[1]1 , λ

[1]2 }; λ

[1]1 = λ(1,0), λ

[1]2 = λ(0,1)

Λ(2) = {λ[2]1 }; λ

[2]1 = λ(2,0)

Actually, the functionals λα are the evaluation func-tionals, δθ, θ ∈ Θ and it is their order in blocks thatwe determine using the constructive method of Newtonbasis.

The sets of index are :I0 = {(0, 0)};I1 = {(0, 0), (1, 0), (0, 1)};I2 = {(0, 0), (1, 0), (0, 1), (2, 0)}.We start with basis gi, i = 1, . . . , 4, and obtain theNewton basis:p[0]1 (x, y) = 1;

p[1]1 (x, y) =

y

b;

p[1]2 (x, y) = − y

2b− x− a

2a;

p[2]1 (x, y) = p

[2]1 (x, y) =

d1x2 + d2y

2

2(d2b2 − d1a2)− y

2b−

− d1a2

2(d2b2 − d1a2).

RΛ,λ(f) = d3[Λ(0), Λ(1), Λ(2), λ; f ]J0 = {(0, 0)}; J1 = {(1, 0), (0, 1)}; J2 = {(2, 0)}d0[λ; f ] = λ(f)d1[Λ(0), λ; f ] = λ(f)− f(a, 0) · λ(p[0]

1 )d2[Λ(0), Λ(1), λ; f ] == d1[Λ(0), λ; f ]− d1[Λ(0), δ(0,b); f ] · λ(p[1]

1 )−−d1[Λ(0), δ(−a,0); f ] · λ(p[1]

2 )d3[Λ(0), Λ(1),Λ(2), λ; f ] = d2[Λ(0),Λ(1), λ; f ]−

−d2[Λ(0),Λ(1), δ(0,−b); f ] · λ(p[2]1 )

References

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