Multivariate Hermite–Birkhoff interpolation by a class of cardinal basis functions

Applied Mathematics and Computation 218 (2012) 9248–9260

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Multivariate Hermite–Birkhoff interpolation by a class of cardinal basis functions Giampietro Allasia, Cesare Bracco ⇑ Department of Mathematics, ‘‘G. Peano’’ – University of Turin, Via Carlo Alberto 10, 10123 Turin, Italy

a r t i c l e

i n f o

Keywords: Multivariate interpolation Arbitrarily distributed data Cardinal basis functions Hermite–Birkhoff interpolation

a b s t r a c t A class of cardinal basis functions for Hermite–Birkhoff interpolation to multivariate real functions on scattered data is constructed. The argument is developed first recalling some classical approaches to the multivariate Hermite interpolation problem, and then introducing suitable cardinal basis functions satisfying a vanishing property on the derivatives. A noteworthy special case involving Shepard’s functions is finally discussed, including some numerical examples. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Let us consider an open and bounded subset X  Rd ; d P 1, and a set of interpolation nodes fx1 ; . . . ; xn g  X. We assume that the nodes are arbitrarily distributed or scattered, i.e. they bear no regular structure at all. The Hermite interpolation problem on scattered data consists of determining a continuous function F : X ! R, belonging to a given interpolation space V, such that its values and derivatives at each interpolation node match certain given values, called data values (see Section 2). Very often the data values are the values assumed at the nodes by the values and derivatives of an unknown continuous function f : X ! R to be interpolated. The Birkhoff interpolation problem on scattered data is a generalization of the Hermite one, which differs only for the way of selecting the derivatives to be interpolated. In the univariate case, that is d ¼ 1, the polynomial interpolation provides the classical Hermite formula (5) (see Section 2). In this case the problem is regular, that is, if the number of nodes and the orders of the derivatives to be interpolated are given, the interpolation polynomial exists and is unique for any set of nodes. On the other hand, it is well known that moving to the multivariate case many troubles arise in the polynomial approach. Actually, it can be proved that, in the multivariate case, the only regular polynomial Hermite interpolation occurs when we have a single node (see [16]). It is then natural to consider nonpolynomial interpolation. Moreover, the Mairhuber–Curtis theorem (see, e.g. [9]) states that a multivariate Lagrange interpolation problem cannot be regular if the interpolation space does not depend on the nodes. Therefore, to solve the Hermite interpolation we must necessarily consider interpolation spaces depending on node locations. A widely used approach involves conditionally positive definite functions, and in particular radial basis functions (see e.g. [10,14,20,21]). In this framework, the interpolation problem is essentially reduced to solving a linear system; in the literature, this brought to a deep study of the conditions to be fulfilled by radial basis functions in order to obtain existence and uniqueness of the solution. On the contrary, in order to get a solution to the Hermite–Birkhoff interpolation problem, the approach presented in this paper uses biorthonormal functions, without solving any linear system. This goal is achieved by introducing a particular class of cardinal basis functions, which, suitably combined with polynomials, give biorthonormal basis functions depending on the

⇑ Corresponding author. E-mail addresses: [email protected] (G. Allasia), [email protected] (C. Bracco). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.002

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node distribution. Shepard’s functions, which belong to the class we will consider, give hints to characterize the class itself. In fact, some vanishing properties of the derivatives of Shepard’s functions (see [19]) were first observed by Shepard himself and by Gordon and Wixom [13], and later developed mainly by Barnhill et al. [5], which suggested the possibility to exploit them in order to get bivariate Hermite interpolants. In the preliminary report [4] it has been sketched that such properties hold not only for the bivariate version of Shepard’s functions, and that they can be used to get Hermite–Birkhoff interpolants. In this paper we will furtherly generalize these ideas, so to get a constructive theorem (Theorem 4.1) which characterizes a class of cardinal basis functions replicating the vanishing properties of the derivatives of Shepard’s functions. We will use Shepard’s functions to provide a few examples and numerical tests. Moreover, we will show, by studying the orders of convergence, that such interpolants also have good approximation properties. This paper is organized as follows. Section 2 deals with some basic ideas of the general interpolation problem, focusing on the biorthonormality property of the well-known polynomial Hermite–Birkhoff interpolation in R. Then, the multivariate case is considered, briefly presenting both the polynomial and the radial basis approaches. In Section 3 cardinal basis interpolation operators are introduced, including Cheney’s construction of cardinal basis functions. The following step, in Section 4, is to fit the structure of those operators to the Hermite–Birkhoff problem. In particular, Theorem 4.1 characterizes a wide class of cardinal basis functions which can be used to reach the aim: the crucial point is the vanishing of some derivatives. Then, combining the considered cardinal basis functions with polynomial factors, we get the nonpolynomial Hermite–Birkhoff interpolation operators in a form very similar to the classical ones in R. Section 5 is devoted to a careful discussion of a noteworthy example of cardinal basis functions related to Shepard’s formula, which are suitable for the construction of Hermite–Birkhoff interpolants. In particular, the significant possibility of getting rational interpolants is pointed out. Finally, Section 6 deals with a localized version of the developed interpolants, the convergence orders, and some numerical examples.

2. Biorthonormality property in interpolation problems All special cases of the interpolation problem we consider fit in well with the following general setting. Let V be a real linear space of dimension n (interpolation space) and let L1 ; L2 ; . . . ; Ln be linear functionals defined on V. For a given set of real values f1 ; f2 ; . . . ; fn we look for an element v 2 V, such that

Li ðv Þ ¼ fi ;

1 6 i 6 n:

ð1Þ

v 1 ; v 2 ; . . . ; v n are the elements of a basis of V, then the solution v can be represented v ¼ a1 v 1 þ a2 v 2 þ    þ an v n , and the interpolation requirement for v takes the form If

Li ðv Þ ¼ a1 Li ðv 1 Þ þ a2 Li ðv 2 Þ þ    þ an Li ðv n Þ ¼ fi ;

as a unique linear combination

1 6 i 6 n:

Hence, a unique solution of (1) exists if and only if the generalized Gram determinant

 n det Li ðv j Þ i;j¼1 – 0:

ð2Þ

If this case, the interpolation problem is said regular. A very interesting case is illustrated by the following result [9]. Theorem 2.1. Let V be a linear space of dimension n and let L1 ; L2 ; . . . ; Ln be independent functionals in the dual space V  . Then, there are n uniquely determined independent elements v 1 ; v 2 ; . . . ; v n of V such that

Li ðv j Þ ¼ dij ;

ð3Þ Pn

Pn

where dij is the Kronecker delta. For any v 2 V we have v ¼ i¼1 Li ðv Þv For every choice of f1 ; f2 ; . . . ; fn , the element v ¼ i¼1 fi v i is the unique solution of the interpolation problem (1). If (3) is satisfied, the elements v j are called biorthonormal with respect to the Li . Then, Theorem 2.1 states that, under the assumption of independence of the functionals, there always exists a biorthonormal basis of the interpolation space V. More importantly, we can observe that using a biorthonormal basis directly proves the existence and uniqueness of the solution, since it implies (2). Therefore, selecting a basis satisfying the biorthonormality property is a suitable approach in order to obtain the regularity of the problem. One of the most common cases of interpolation occurs when V is spanned by some real valued functions belonging to Cq ðXÞ; X  R; q 2 N0 , and the functionals are  i.

k

Lhk ðf Þ ¼

d f ðxh Þ k

dx

;

16h6n

for distinct points x1 ; x2 ; . . . ; xn 2 X, and for some 0 6 k 6 q. Let us give a noteworthy example. Let a1 ; a2 ; . . . ; an be integers P 1; m ¼ a1 þ a2 þ    þ an  1, and V be the space spanned by the polynomials of degree m

g ij ðxÞ ¼

 aX s  i j1 1 XðxÞ 1 d ðx  xi Þai ðx  xi Þs ; j! ðx  xi Þai j s¼0 s! dxs XðxÞ x¼xi

ð4Þ

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with XðxÞ ¼ ðx  x1 Þa1 ðx  x2 Þa2    ðx  xn Þan . If we consider the functionals Lh;k for h ¼ 1; 2; . . . ; n, and 0 6 k 6 ah  1, then we get the univariate polynomial Hermite interpolation (see e.g. [6]). The existence of a unique solution, and therefore the regularity of the problem, is an immediate consequence of the biorthonormality of the basis functions g i;j with respect to the functionals, that is, we have

Lhk ðg ij Þ ¼ dih djk : Hence, given a set of real values fij , for i ¼ 1; 2; . . . ; n, and 0 6 j 6 ai  1, the polynomial n a i 1 X X

pm ðxÞ ¼

fij g ij ðxÞ

ð5Þ

j¼0

i¼1

is the unique element of P1m for which

Lhk ðpm Þ ¼ fhk ;

0 6 k 6 ah  1:

1 6 h 6 n;

Here and in the following fhk can be interpreted as the derivative of order k at the point xh of an underlying function f. Birkhoff interpolation (see e.g. [15]) is different from the Hermite one because gaps in the derivatives to be interpolated are allowed, that is, for each 0 6 h 6 n the index k takes values in Dh , a given finite subset of N0 . For h ¼ 1; 2; . . . ; n and Dh ¼ fk 2 N0 jk 6 ai  1g, we get the Hermite interpolation. In the multivariate case the situation is more complicated and we have different types of Hermite interpolation (see e.g. [16]). In a case, given a set of distinct points fxi : 1 6 i 6 ng in X  Rd , let the interpolation functionals be

Libi ðf Þ ¼ Dbi f ðxi Þ 

@ jbi j f ðxi Þ b

b b @t 1i1 @t 2i2

   @t did

ð6Þ

;

where bi ¼ ðbi1 ; bi2 . . . ; bid Þ 2 Nd0 ; 1 6 i 6 n, and j bi j¼ bi1 þ bi2 þ    þ bid . Thus, given a functional space V  Cq ðXÞ and n nonnegative integers k1 ; . . . ; kn with ki 6 q for i ¼ 1; . . . ; n, the Hermite interpolation of total degree is the problem of finding p 2 V satisfying

Libi ðpÞ ¼ fibi ;

0 6j bi j6 ki ;

i ¼ 1; 2; . . . ; n

for arbitrary real values fibi .     In another case, if we take n d-tuples ki ¼ ðki1 ; ki2 ; . . . ; kid Þ 2 Nd0 ; 1 6 i 6 n, we can define the Hermite interpolation of coordinate degree, consisting in finding p 2 V such that 

Libi ðpÞ ¼ fibi ;

0 6 bih 6 kih ;

h ¼ 1; 2; . . . ; d;

i ¼ 1; 2; . . . ; n

for arbitrary real values fibi . Finally, given n finite subsets Di  Nd0 , with maxb2Di j b j6 q for 1 6 i 6 n, the multivariate Birkhoff interpolation (see, e.g. [15]) consists in finding p 2 V such that

Libi ðpÞ ¼ fibi ;

bi 2 Di ;

i ¼ 1; 2; . . . ; n

for arbitrary real values fibi . However, in contrast with the univariate case, the regularity of the problem can be obtained only for very particular cases, that is, when the node set S contains a single element (see, e.g. [16,7]). Let x1 2 Rd and k1 2 N0 , the Taylor–Hermite interpolation problem of total degree at the node x1 consists in finding p belonging to the polynomial space

( d k1

P ¼

)

X

c

ac x : c 2

Nd0 ;

d

x 2 R ; aa 2 R

jcj6k1

and such that:

Db1 pðx1 Þ ¼ f1b1 ;

b1 ¼ ðb11 ; . . . ; b1d Þ 2 Nd0 : jb1 j 6 k1

for any arbitrary real value f1b1 . The problem is regular for any choice of x1 , because we have as many conditions as the dimension of Pdk1 and

Db1



ðx  x1 Þa a!



¼ db1 a ;

ð7Þ

x¼x1

where a! ¼ a1 !    ad ! and ðx  x1 Þb1 ¼ ðx  x1 Þb11    ðx  x1 Þb1d . The solution is the polynomial

Tðx; f ; x1 ; k1 Þ ¼

X jaj6k1

d

 k1

f1a

ðx  x1 Þa : a!

Given x1 2 R and 2 Nd0 , the Taylor–Hermite interpolation problem of coordinate degree at the node x1 consists in finding p belonging to the polynomial space

G. Allasia, C. Bracco / Applied Mathematics and Computation 218 (2012) 9248–9260

Pdðk11 ;...;k1d Þ ¼

8 > > < X > > :06ch 6k1h ;

ac xc : c 2 Nd0 ; x 2 Rd ; aa 2 R

16h6d

9251

9 > > = > > ;

and such that: 

b1 ¼ ðb11 ; . . . ; b1d Þ 2 Nd0 : b1h 6 k1h ;

Db1 pðx1 Þ ¼ f1b1 ;

h ¼ 1; . . . ; d

for any arbitrary real value f1b1 . The problem is regular for any choice of x1 , because the number of conditions matches the dimension of Pdðk11 ;...;k Þ and (7) holds. The solution is the polynomial 1d

 Cðx; f ; x1 ; k1 Þ

X

¼

f1a

06ah 6k1h ; 16h6d

ðx  x1 Þa : a!

Let x1 2 Rd and let D1 be a finite subset of Nd0 ; the Taylor–Birkhoff interpolation problem at the node x1 consists in finding p belonging to the polynomial space

(

PdD1 ¼

X

)

ac xc : c 2 Nd0 ; x 2 Rd ; aa 2 R

c2D1

and such that:

Db1 pðx1 Þ ¼ f1b1 ;

b1 2 D1

for any given real value f1b1 . The problem is regular for any choice of x1 , because we have as many conditions as the dimension of PdD1 and (7) holds. The solution is the polynomial

Bðx; f ; x1 ; D1 Þ ¼

X

f1;a

a2D1

ðx  x1 Þa : a!

In the literature, a widely used and effective approach for multivariate Hermite–Birkhoff interpolation is based on radial basis functions. Let us consider an interpolant of the form

FðxÞ ¼

n X

ai Uðx  xi Þ þ

i¼1

q X

bk pk ðxÞ;

x 2 X;

ð8Þ

k¼1

where U is a radial function and q is the dimension of a suitable polynomial space with basis p1 ; . . . ; pq . The coefficients ai and bk are determined by solving the system

Db Fðxi Þ ¼ Db f ðxi Þ;

b 2 Di ;

i ¼ 1; 2; . . . ; n:

This leads to a deep study of the dependence of the solvability of this system on the properties of the radial function U (see, e.g. [14,21]). While this approach has been successfully developed in literature and gave rise to several efficient interpolation methods, in this paper, as already mentioned, we solve instead the Hermite–Birkhoff interpolation problem by constructing suitable biorthonormal functions g~ja , that is, satisfying

Libi ðg~ja Þ ¼ dij dbi a ;

a 2 Di :

i; . . . ; n;

As a consequence, we can then consider an interpolant of the type

FðxÞ ¼

n X X

g~ia :

i¼1 a2Di

In the next two sections, such biorthonormal functions g~ja are constructed by a class of cardinal basis functions. 3. Cardinal basis functions We are now interested in considering a family of cardinal basis functions, which provides a regular Lagrange interpolation for any choice of the node set, and are basic for the construction of the biorthonormal functions we will employ in the definition of Hermite–Birkhoff interpolants in Section 4 (for a deeper discussion of cardinal basis functions see, e.g. [1,3,10]). Definition 3.1. Let S ¼ fxi ; 1 6 i 6 ng be a set of distinct points, in general arbitrarily distributed in a bounded domain

X  Rd ; d P 1, with associated real values ffi ; 1 6 i 6 ng, and let VðXÞ be a linear space, spanned by cardinal basis functions g j : X ! R; 1 6 j 6 n, such that n X g j 2 C 0 ðXÞ; g j ðxÞ P 0; g j ðxÞ ¼ 1; g j ðxi Þ ¼ dji ; ð9Þ j¼1

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where dji is the Kronecker delta. Then, a Lagrange cardinal basis interpolant in VðXÞ is

Fðx; f ; SÞ ¼

n X

fj g j ðxÞ;

x 2 X:

ð10Þ

j¼1

The interpolation functionals we consider here are for all f 2 V

Li ðf Þ ¼ f ðxi Þ;

16i6n

and therefore the basis functions g j ; 1 6 j 6 n, are biorthonormal with respect to the functionals by definition. The interpolation problem of finding F 2 VðXÞ such that

Fðxi Þ ¼ fi ;

16i6n

is then regular and has the unique solution Fðx; f ; SÞ. We remark on a few noteworthy properties: Property 3.2. The Lagrange cardinal basis interpolant Fðx; f ; SÞ enjoys the characteristic properties of a weighted arithmetic mean: 1. minj fj 6 Fðx; f ; SÞ 6 maxj fj ; 2. if fj ¼ const; 1 6 j 6 n, then Fðx; f ; SÞ  const; 3. Fðx; f ; SÞ, for each x fixed, is the unique solution of the weighted least squares problem

min

f 2VðXÞ

n X 

2 f ðxÞ  fj g j ðxÞ:

ð11Þ

j¼1

Solving a problem like (11), but with weights wj ðxÞ; 1 6 j 6 n, by the moving weighted least squares method gives

Fðx; f ; SÞ ¼

n X j¼1

wj ðxÞ : fj Pn k¼1 wk ðxÞ

ð12Þ

Hence, if we take

wj 2 C 0 ðXÞ;

wj ðxÞ P 0;

wj ðxi Þ ¼ 0 for j – i;

wj ðxj Þ > 0;

then we can set

wj ðxÞ ; g j ðxÞ ¼ Pn k¼1 wk ðxÞ

16j6n

ð13Þ

and these g j ðxÞ can be interpreted as the basis functions in Definition 3.1. An important way of constructing the weights wj ðxÞ is the following one suggested by Cheney ([8], see also [2]). Let aðx; yÞ be a real function continuous on X  X such that aðx; xi Þ > 0, if x – xi , and aðxi ; xi Þ ¼ 0; for all x 2 X and xi 2 S. Then, setting n Y

wj ðxÞ ¼

aðx; xi Þ;

ð14Þ

i¼1;i–j

we get

Qn

aðx; xi Þ ; i¼1;i–k aðx; xi Þ

i¼1;i–j

g j ðxÞ ¼ Pn Qn k¼1

ð15Þ

and

Fðx; f ; SÞ ¼

n X j¼1

fj g j ðxÞ ¼

n X j¼1

Qn

aðx; xi Þ ; i¼1;i–k aðx; xi Þ

i¼1;i–j

fj Pn Qn k¼1

ð16Þ

or equivalently

Fðx; f ; SÞ ¼

n X j¼1

1=aðx; xj Þ ; fj Pn k¼1 1=aðx; xk Þ

Fðxi ; f ; SÞ ¼ f ðxi Þ;

1 6 i 6 n:

ð17Þ

As pointed out in [3], the cardinal basis functions (9), which also include functions which cannot be obtained by Cheney’s construction, can be used to obtain Lagrange interpolants (10). In particular, FðxÞ is a positive linear operator, i.e. a Korovkin-type operator, which satisfies not only interpolation properties, but also the approximation properties presented in [3]. However, in this paper we are interested in a specific class of cardinal basis functions, which is suitable to construct Hermite–Birkhoff interpolants and will be introduced in the next section.

G. Allasia, C. Bracco / Applied Mathematics and Computation 218 (2012) 9248–9260

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4. A class of Hermite–Birkhoff interpolants on scattered data We will now consider a particular class of the cardinal basis functions introduced in Section 3, satisfying a vanishing property about the derivatives, which are useful in order to construct biorthonormal functions suitable to get nonpolyomial Hermite–Birkhoff interpolants. Given an open and bounded domain X  Rd (preferably convex) and a set of n distinct scattered points S  X, let us consider a set of cardinal basis functions g 1 ðxÞ; g 2 ðxÞ; . . . ; g n ðxÞ belonging to Cq ðXÞ for q 2 N0 and satisfying the additional property

Dbi g j ðxi Þ ¼



dij ; if j bi j¼ 0; 0;

ð18Þ

if 0
for i; j ¼ 1; 2; . . . ; n. While it has already been suggested, e.g. in [4], that property (18) can be enough to have cardinal basis functions suitable to Hermite–Birkhoff interpolants, it is clear that it would be very desirable to have an actual method to construct such functions, in a similar way to Cheney’s construction for the general cardinal basis functions. The following constructive theorem characterizes a class of cardinal basis functions satisfying property (18). Theorem 4.1. Let aðx; yÞ be a real function, continuous and bounded on X  X, such that aðx; xi Þ > 0 for all x 2 X and x – xi . Moreover, let aðx; xi Þ 2 C q ðXÞ for q 2 N0 , with derivatives bounded on X up to the qth order, and such that

½Dbi aðx; xi Þx¼xi ¼ 0;

i ¼ 1; 2; . . . ; n;

0 6 jbi j 6 q:

ð19Þ

The corresponding cardinal basis functions

Qn

aðx; xi Þ ; j ¼ 1; 2; . . . ; n; i¼1;i–k aðx; xi Þ

i¼1;i–j

g j ðxÞ ¼ Pn Qn k¼1

ð20Þ

belong to C q ðXÞ and satisfy (18). The proof of this theorem is given in Appendix A. We introduce now the functions

g ja ðxÞ ¼

ðx  xj Þa g j ðxÞ; a!

j ¼ 1; 2; . . . ; n;

a 2 Nd0 ; 0 6 jaj 6 q;

ð21Þ

which can play the same role as the polynomials (4) do in the univariate Hermite interpolation, in the sense that they enjoy the biorthonormality property

Libi ðg ja Þ ¼ Dbi g ja ðxi Þ ¼ dij dbi a ;

i; j ¼ 1; 2; . . . ; n;

0 6j a j; j bi j6 q:

ð22Þ

So, given f 2 Cq ðXÞ, we can state that the interpolants

F T ðx; f ; SÞ ¼

n X X

Da f ðxj Þg ja ðxÞ;

j¼1 06jaj6kj

F C ðx; f ; SÞ ¼

F B ðx; f ; SÞ ¼

n X

X

j¼1

06jah j6kjh

n X X

Da f ðxj Þg ja ðxÞ;

ð23Þ

Da f ðxj Þg ja ðxÞ

j¼1 a2Dj

are solutions to the Hermite interpolation problem of total degree, to the Hermite interpolation problem of coordinate degree, and to the Birkhoff interpolation problem, respectively. The biorthonormality property (22) is easily proved since, by Theorem 4.1, we have

Libi ðg ja Þ ¼



bid bi1 bi2 X X bi2 b 1 X bi1    id  Dm1 ;m2 ;...;md ½ðxi  xj Þa Dðbi1 m1 Þ;ðbi2 m2 Þ;...;ðbid md Þ g j ðxi Þ  a! m ¼0 m ¼0 m ¼0 m1 m2 md 1

2

d

1 ¼ Dbi1 ;bi2 ;...;bid ½ðxi  xj Þa dij ¼ dabi dij : a! From (22) it immediately follows that each of these solutions is unique in the space spanned by the g ja involved in the expression of the interpolant. The interpolants in (23) can also be written in the form

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F T ðx; f ; SÞ ¼

n X

Tðx; f ; xj ; kj Þg j ðxÞ;

ð24Þ

j¼1

F C ðx; f ; SÞ ¼

n X



Cðx; f ; xj ; kj Þg j ðxÞ;

ð25Þ

Bðx; f ; xj ; Dj Þg j ðxÞ;

ð26Þ

j¼1

F B ðx; f ; SÞ ¼

n X j¼1



where Tðx; f ; xj ; kj Þ; Cðx; f ; xj ; kj Þ, and Bðx; f ; xj ; Dj Þ are the Taylor–Hermite polynomial of total degree kj at the node xj , the Tay  lor–Hermite polynomial of coordinate degree kj1 ; . . . ; kjd at the node xj , and the Taylor–Birkhoff polynomial with orders of the interpolated derivatives b 2 Dj at the node xj , respectively. Therefore the class of cardinal functions g j , which has been characterized by the Theorem 4.1, can be used to construct a class of Hermite–Birkhoff nonpolynomial interpolants. This has been essentially done by combining the g j with the suitable Taylor-type polynomials, as shown by (24)–(26). The vanishing of the derivatives of the g j by Theorem 4.1 and the regularity of the Taylor-type interpolation give as a result that the basis of functions g ja , depends on the distribution of the nodes. The biorthonormal property (22) enjoyed by the new basis implies the regularity of the problem, as desired. Note that the Lagrange cardinal basis interpolants presented in Section 3 and in [3] are particular cases of these Hermite–  Birkhoff interpolants: in fact, they can be obtained from (24)–(26) setting, respectively, kj ¼ 0; kj ¼ ð0; . . . ; 0Þ and Dj ¼ fð0; . . . ; 0Þg, for j ¼ 1; . . . ; n. The interpolation operators (24)–(26) also satisfy some noteworthy reproduction properties. If p is a polynomial of total degree 6 mini ki , then

F T ðx; p; SÞ ¼ pðxÞ;

ð27Þ

that is, the polynomials of total degree 6 mini ki are reproduced by the operator F T . In fact, recalling that Tðx; p; xj ; kj Þ ¼ pðxÞ if p is a polynomial of total degree 6 kj , and that the g i ; i ¼ 1; . . . ; n, form a partition of unity, we have

F T ðx; p; SÞ ¼

n X

Tðx; p; xj ; kj Þg j ðxÞ ¼

j¼1

n X

pðxÞg j ðxÞ ¼ pðxÞ:

j¼1

Analogous properties are enjoyed by F B and F C . 5. A constructive example: Shepard’s operator Cardinal basis functions of the type considered in Theorem 4.1 can be constructed for instance by setting

aðx; xi Þ ¼ dl ðx; xi Þ, where dðx; xi Þ is the Euclidean distance between x and xi , and l 2 Rþ . l

Theorem 5.1. The function aðx; yÞ ¼ d ðx; xi Þ, with l > 1, satisfies the hypothesis of Theorem 4.1 for q ¼ l  1 if l is an integer, and for q ¼ dle if l is not an integer, where dle is the largest integer
Qn

l

i¼1;i–j d

g j ðxÞ ¼ Pn Qn k¼1

ðx; xi Þ l

i¼1;i–k d

ðx; xi Þ

;

0 6 j 6 n:

ð28Þ

By Theorem 5.1, it is clear that Shepard’s functions belong to the class of functions (20), and therefore satisfy the property (18) on derivatives. Comments about the vanishing of the derivatives of Shepard’s functions were already made in some classic papers by Shepard himself [19], Schumaker [18], and Gordon and Wixom [13]. These observations have been later developed by Barnhill et al. [5], which proved that Shepard’s functions enjoy property (18) in the bivariate case d ¼ 2. Actually, in constructing the class of functions (20), one of our goal has been replicating on a more general scheme the behavior of the derivatives of Shepard’s functions, since it is a crucial point in order to use them for Hermite–Birkhoff interpolation. The continuity class of the g j defined in (28), and hence the continuity class of the interpolants (24)–(26) too, depends upon l: (i) if l is an even integer, then g j 2 C 1 ðXÞ, (ii) if l is an odd integer, then g j 2 C l1 ðXÞ, and (iii) if l is not an integer, then g j 2 C dle ðXÞ. Note that if l is an even integer, then each cardinal basis functions g j ðxÞ is a ratio of polynomials. Hence, also the corresponding functions g j;a defined in (21) are ratio of polynomials, being products of g j and a polynomial, and so the interpolants (23) are rational. Combining Theorems 4.1 and 5.1 it is clear that, choosing a suitable l, it is always possible, at least from a theoretical point of view, to use Shepard’s cardinal basis functions to solve the Hermite–Birkhoff interpolation problem involving derivatives of any order. For example, taking ki ¼ 1; i ¼ 1; . . . ; n, and l ¼ 2, we have

G. Allasia, C. Bracco / Applied Mathematics and Computation 218 (2012) 9248–9260

Tðx; f ; xi ; 1Þ ¼ f ðxi Þ þ

9255

d X @f ðxi Þ ðt h  tih Þ: @t h h¼1

Then, the interpolation operator (24) appears

F T ðx; f ; SÞ ¼

n X j¼1

Qn

l

i¼1;i–j d

Tðx; f ; xi ; 1Þ Pn Qn k¼1

ðx; xi Þ l

i¼1;i–k d

ðx; xi Þ

;

which interpolates the function f and its first derivatives at the nodes xi ; 1 6 i 6 n. 6. Localizing scheme and approximation performances In Section 4, we characterized a class of cardinal basis functions g j suitable to be combined with incomplete Taylor expansions in order to get nonpolynomial Hermite–Birkhoff interpolants. Then, in Section 5, we have shown an example of functions belonging to this class, namely Shepard’s functions, and we have constructed interpolants which are solutions to the Hermite–Birkhoff problem in the space generated by the corresponding functions g ja (21) (see also [4] for this particular case). The theoretical solution to the Hermite–Birkhoff interpolation problem, which is the main goal of the present work, is in itself a relevant topic, widely considered in the literature. Nevertheless the developed tools do not play just a theoretical role, but actually they can be adapted for numerical application purposes. In particular, the constructive example involving Shepard’s functions can be used to carry out numerical experiments and can be suitably modified in order to meet application requirements. So, going a little beyond the aim of the paper, we devote this section to the discussion of a few important application-oriented aspects of the defined interpolants and to the study of the orders of convergence of the interpolants (which were not touched in the particular case presented in [4]). Applying the interpolants (24)–(26), in many cases it could be convenient to use a localized version of Cheney’s construction, at least for computational reasons. More precisely, we consider the operators

F~T ðx; f ; SÞ ¼

n X

Tðx; f ; xj ; kj Þg~j ðxÞ;

ð29Þ

Cðx; f ; xj ; kj Þg~j ðxÞ;

ð30Þ

Bðx; f ; xj ; Dj Þg~j ðxÞ;

ð31Þ

j¼1

F~C ðx; f ; SÞ ¼

n X j¼1

F~B ðx; f ; SÞ ¼

n X j¼1

where

si ðxÞð1=aðx; xi ÞÞ ; s aðx; xk ÞÞ

g~i ðxÞ ¼ Pn

ð32Þ

k¼1 k ðxÞð1=

with

 Þ, such that si : X ! Rþ0 ; si 2 Cq ðX

sðxÞ > 0 for x : dðx; xi Þ < d; sðxÞ ¼ 0; for x : dðx; xi Þ P d and d > 0 is a suitably chosen radius. Hence the interpolants, when evaluated at any x 2 X, consider only the nodes closest to x, that is, the nodes xi such that dðx; xi Þ < d, which allows, with respect to the non-localized interpolants in (24) and the basic particular studied in [4], a noteworthy cut in the number of operations needed. We note that (29)–(31) are continuous operators like (24)–(26). We also note that the cardinal basis functions in (32) are still a partition of unity, and therefore the reproduction properties (27) still holds for the localized interpolants. Finally, it can be easily proved, by arguments analogous to the ones used for the proof of Theorem 4.1, that the functions g~i ; i ¼ 1; . . . ; n, enjoy the biorthonormality property (18). The localization can be performed by taking, for instance,



si ðxÞ ¼ 1  d2 ðx; xi Þ=d2

qþ1 þ

;

x 2 X;

i ¼ 1; . . . ; n;

so to get the cardinal basis functions

 qþ1 2 1  d ðx; xi Þ=d2 =aðx; xi Þ þ ; g~i ðxÞ ¼  qþ1 Pn 2 2 =aðx; xk Þ k¼1 1  d ðx; xk Þ=d

d > 0:

þ



Note that setting in (29)–(31) setting, respectively, kj ¼ 0; kj ¼ ð0; . . . ; 0Þ and Dj ¼ fð0; . . . ; 0Þg, for j ¼ 1; . . . ; n, we get a localized version of the Lagrange cardinal basis interpolants presented in Section 3 and in [3]. The localized versions of the interpolants also give us the opportunity to obtain error estimates involving the so-called fill distance, defined by

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G. Allasia, C. Bracco / Applied Mathematics and Computation 218 (2012) 9248–9260

hS;X :¼ sup min dðx  xi Þ: xi 2S

x2X

Let us consider the Birkhoff interpolant (31), since both (29) and (30) can be formally reduced to particular cases of (31). Let



Rj ¼ a 2 Dj : 8b 2 Nd0 with j b j6j a j; then b 2 Dj



and

  m ¼ min max j a j: a 2 Rj : 16j6n

Note that m is defined such that each Taylor-type expansion Bðx; f ; xj ; Dj Þ can be seen as a complete Taylor expansion up to order m plus other terms of higher degree. For any f : X ! R with f 2 Cq ðXÞ and for any x 2 X, we have



P n P

n ~ ~ ~ j f ðxÞ  F B ðx; f ; SÞ j¼ f ðxÞg j ðxÞ  Bðx; f ; xj ; Dj Þg j ðxÞ

j¼1 j¼1

P P

n 6 ½f ðxÞ  Bðx; f ; xj ; Dj Þg~j ðxÞ j¼

f ðxÞ  Bðx; f ; xj ; Dj Þ j g~j ðxÞ;

j¼1 j:dðx;xj Þ
ð33Þ

since the cardinal basis functions g~j are a partition of unity, and each g j is non-zero only inside the ball of radius d centered at xi . Now, since each Bðx; f ; xj ; Dj Þ is a Taylor expansion complete up to the mth order, we can use the estimate mþ1

j f ðxÞ  Bðx; f ; xj ; Dj Þ j6 C j d

ðx  xj Þ;

C j 2 Rþ :

ð34Þ

Using (34) and exploiting again the partition of unity property, by (33) we get

j f ðxÞ  F~B ðx; f ; SÞ j6

X

mþ1

Cjd

ðx; xi Þg~j ðxÞ 6 dmþ1

j:dðx;xj Þ
X j:dðx;xj Þ
C j g~j ðxÞ 6 Cdmþ1 ;

C ¼ max C j j

since the js considered in the sum are such that j x  xj j< d. Moreover, taking d ¼ KhS;X with K P 1, we obtain the estimate mþ1 j f ðxÞ  F~B ðx; f ; SÞ j6 CKhS;X ;

ð35Þ mþ1 OðhS;X Þ.

which implies that the method has approximation order In particular, the interpolant (29) has approximation order min k OðhS;X i i Þ. This error bound clearly shows that the introduction of the Taylor-type expansions in the cardinal basis interpolation scheme gives higher approximation orders. In particular, this means that, considering Taylor expansions as local approximants instead of the simple function values f ðxi Þ, as in the (localized versions of) the classical Shepard’s interpolant and of the Lagrange cardinal basis interpolants presented in Section 3 and in [3], significantly improves the approximation performances. Moreover, note that, unlike in the study of convergence of Lagrange cardinal basis interpolants in [3], here we have established a relationship between the error and the fill distance associated to the node set S. Let us consider a numerical example in order to show the just mentioned improvement of performance. We consider Franke’s test function (see [12])

f ðt 1 ; t 2 Þ ¼ 0:75 exp½0:25ð9t 1  2Þ2  0:25ð9t 2  2Þ2  þ 0:75 exp½ð9t 1 þ 1Þ2 =49  ð9t2 þ 1Þ=10 þ 0:5 exp½0:25ð9t 1  7Þ2  0:25ð9t2  3Þ2   0:2 exp½ð9t 1  4Þ2  ð9t 2  7Þ2 : This reliable function, commonly used to test surface approximation methods, is considered here to give merely an example, since several tests have been done considering other functions and the obtained results are comparable. First, we tested the classical Shepard’s interpolant in localized form

~ f ; SÞ ¼ Fðx;

n X

f ðxj Þg~j ðxÞ;

ð36Þ

j¼1

where the cardinal basis functions are

 3 2 3 1  d ðx; xi Þ=d2 ð1=d ðx; xj ÞÞ þ ; g~j ðxÞ ¼  3 Pn 2 3 2 ð1=d ðx; xk ÞÞ k¼1 1  d ðx; xi Þ=d

d ¼ 2hS;X :

ð37Þ

þ

Then, we compare the formula (36) with the localized Hermite interpolant (29) using Taylor expansions of total degree 2, that is

F~T ðx; f ; SÞ ¼

n X

Tðx; f ; xj ; 2Þg~j ðxÞ;

ð38Þ

j¼1

~j ðzÞ are the same as in (37). The interpolation nodes are obtained dividing the square where the cardinal basis function g ½0; 1  ½0; 1 in a regular r  r grid (r ¼ 10; 20; 30; 40; 50; 60; 70; 80; 90; 100), and the interpolant is evaluated on a regular

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G. Allasia, C. Bracco / Applied Mathematics and Computation 218 (2012) 9248–9260 Table 1 Evaluation of Franke’s function by the operators F~ in (36) and F~T in (38) on X ¼ ½0; 1  ½0; 1. # Nodes

Gridded nodes

Halton nodes

F~ RMSE 100

F~T RMSE 2

2:930  10

F~ RMSE

1:429  10

3

F~T RMSE

3:433  10

2

4:174  103

2

7:250  104

400

1:310  10

2:015  10

1:629  10

900

8:139  103

8:124  105

1:004  102

1:757  104

1600

3

6:365  10

4:499  10

5

3

1:082  104

2500

4:578  103

2:488  105

6:018  103

6:500  105

3600

3

4:216  10

1:948  10

5

3

4:496  105

4900

3:576  103

1:424  105

4:288  103

2:917  105

6400

3

2:891  10

9:816  10

6

3

2:442  105

8100

2:788  103

8:511  106

3:401  103

2:026  105

10000

3

6

3

1:539  105

2

4

6:505  10

2:432  10

7:779  10 4:958  10 3:789  10 2:986  10

33  33 grid, in order to get, as long as possible, a reliable picture of the approximation results. Then, we made the same tests using the pseudo-random Halton points with generating primes 2 and 3. The results are reported in Table 1. As suggested by the error bound (35) involving the fill distance, the Hermite interpolant (38) gives much better approximation performances than the classical Shepard’s interpolant (36). However, it should be noted that in real applications the data concerning derivatives are hardly available, and therefore, in general, the Hermite interpolation scheme could not be applied. From this point of view, a practical solution is to get local approximants at the nodes obtained by means of numerical methods, as, e.g., the moving least squares method using weight functions with reduced compact support. This approach has been proposed by Franke and Nelson in 1980 [11] and applied in standard routines, such as that by Renka [17]. 7. Conclusions We have shown that Hermite–Birkhoff interpolation for scattered data in Rd is possible considering nonpolynomial operators expressed as combinations of (incomplete) Taylor expansions and cardinal basis functions. Such interpolants have been obtained by considering a suitable class of cardinal basis functions characterized by Theorem 4.1, which generalizes a vanishing property of the derivatives of Shepard’s functions studied in [4]. The biorthonormality property, which is satisfied by the new basis (21) of functions in Rd , implies the desired regularity of the interpolation problem and allows us to evaluate the solution without solving any linear system. The Hermite and Birkhoff operators in Rd are effectively constructed and are expressed by the formulas (23) or by the different, but equivalent, formulas (24)–(26). A noteworthy example is given, namely, Shepard’s operator, which is important in itself but it is also useful to put in practice our discussion. The localized versions of the interpolants, given in Section 6, are much more convenient to be used in practice and have good approximation properties, as we proved by studying the relationship between the error and the fill distance, which shows that the order of convergence, roughly speaking, increases as the degree of the employed Taylor expansions grows. Acknowledgements The authors thank a referee for his careful reviewing and useful suggestions. Appendix A

Proof of Theorem 4.1. The case j bi j¼ 0 is easily seen, because

Qn

aðxi ; xh Þ ¼ dij ; 1 6 i; j 6 n: h¼1;h–k aðxi ; xh Þ

h¼1;h–j

g j ðxi Þ ¼ Pn Qn k¼1

For 0
Dbi g j ðxi Þ ¼

bi1 X

bi2 X

m1 ¼0 m2 ¼0



bid X md ¼0



bi1 m1



bi2 m2



3 11 2 0

n 7 C BX Y b 6 m1 ;m2 ;...;md Y ðbi1 m1 Þ;ðbi2 m2 Þ;...;ðbid md Þ B    id 6 D aðxi ; xh Þ7 aðxi ; xh ÞC 5D A : @ md 4

h¼1; h–jn

k¼1 h¼1; h–kn

ð39Þ The general term in the sum contains the product of two factors involving derivatives. We will prove that the first factor vanishes, whereas the second factor is bounded.

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G. Allasia, C. Bracco / Applied Mathematics and Computation 218 (2012) 9248–9260

(a) The first factor is

Dm1 ;m2 ;...;md

n Y

aðxi ; xh Þ ¼

m1 X m2 X



r1 ¼0 r 2 ¼0

h¼1; h–j



md X m2 m1 r1

r d ¼0

r2





n Y md  Dr1 ;r2 ;...;rd aðxi ; xi Þ  Dðm1 r1 Þ;ðm2 r2 Þ;...;ðmd rd Þ aðxi ; xh Þ: rd h¼1; h–i;j

This factor vanishes because, by hypothesis,

Dr1 ;r2 ;...;rd aðxi ; xi Þ ¼ 0;

for all i ¼ 1; 2; . . . ; n and 0
This argument does not work for i ¼ j, that is, to show Dbj g j ðxj Þ ¼ 0 , since we cannot isolate the factor Dr1 ;r2 ;...;rd aðxi ; xi Þ; hence, we will discuss this particular case below. (b) The second factor is

Dðbi1 m1 Þ;ðbi2 m2 Þ;...;ðbid md Þ G1 ðxi Þ; where

GðxÞ ¼

n Y n X

aðx; xh Þ:

k¼1 h¼1; h–k

Now GðxÞ is positive and bounded in X. The derivatives of GðxÞ of order less or equal to q, being combinations of derivatives of aðx; xh Þ, are also bounded. Hence, there exist numbers cG > 0 and C G < 1 so that GðxÞ P cG and j Dbi GðxÞ j6 C G for all 0
j D1;0;...;0 G1 ðxÞ j¼j

1 ½G2 ðxÞ

D1;0;...;0 GðxÞ j6

CG : c2G

Now, we suppose that Dbi G1 ðxÞ is bounded for any bi 2 Nd such that j bi j6 k, and we will prove that this is true also for j bi j¼ k þ 1. Note that, since we have

Dbi G2 ðxÞ ¼

bi1 X m1 ¼0





bid X b bi1    id Dm1 ;...;md G1 ðxÞDðbi1 m1 Þ;...;ðbid md Þ G1 ðxÞ; m m 1 d m ¼0 d

the induction hypothesis directly implies that also the derivatives Dbi G2 ðxÞ, for j bi j6 k, are bounded. Then, for any bi 2 Nd ; j bi j¼ k þ 1, we have bi

j D GðxÞ

1

( )



ðbi1 1Þ;bi2 ;...;bid 1 1;0;...;0 D j¼j D fD G ðxÞg j¼ g D GðxÞ 2

G ðxÞ







bP h i bid bi2 P bi2 b bi1  1

i1 1 P ¼    id Dðm1 þ1Þ;m2 ;...;md GðxÞ  Dðbi1 1m1 Þ;ðb12 m2 Þ;...;ðbid md Þ G2 ðxÞ 

m1 ¼0m2 ¼0 md ¼0 m1 m2 md



b bP 1 b i1 i2 id P P bi2 b bi1  1 6    id j Dðm1 þ1Þ;m2 ;...;md GðxÞ j  j Dðbi1 1m1 Þ;ðb12 m2 Þ;...;ðbid md Þ G2 ðxÞ j :  m1 m2 md md ¼0 m1 ¼0m2 ¼0 ðbi1 1Þ;bi2 ;...;bid

1;0;...;0

1

Since j bi j¼ k þ 1, any derivative of G2 ðxÞ in the above expression has order less than or equal to k, and therefore is bounded. Hence, by the induction hypothesis each term is bounded and so the result for i – j follows. Theorem 4.1 is true also for i ¼ j, because we have for all x 2 X

Dbi

" n X

# g j ðxÞ ¼ Dbi 1 ¼ 0;

0 < jbi j < l

j¼1

and then

X

Dbi g j ðxÞ þ Dbi g i ðxÞ ¼ 0:

j¼1; j–i

From this, taking x ¼ xi , it follows Dbi g j ðxi Þ ¼ 0; 0 < jbi j 6 q. To complete the proof, we observe that g j 2 Cq ðXÞ for j ¼ 1; 2; . . . ; n, as it results from the whole discussion on Dbi g j ðxÞ. h

We will now give the proof of Theorem 5.1, which implies that Shepard’s cardinal basis functions belong to the class of functions (20), so that they can be used to construct a particular case of the interpolants (23), providing Hermite–Birkhoff interl polation for any d 2 N. First, we need two lemmas on the derivatives of dðx; xi Þ and d ðx; xi Þ.

G. Allasia, C. Bracco / Applied Mathematics and Computation 218 (2012) 9248–9260

9259

Lemma 8.1. If bi ¼ ðbi1 ; bi2 ; . . . ; bid Þ; bih 2 N0 ; 1 6 i 6 n; 1 6 h 6 d, and

Dbi ¼

@ bi1 þbi2 þþbid b b @t1i1 @t 2i2

b    @t did



@ jbi j b b @t1i1 @t 2i2

b

   @t did

then 1jbi j

Dbi dðx; xi Þ ¼ d

ðx; xi Þpbi ðAi1 ; Ai2 ; . . . ; Aid Þ;

ð40Þ

where pbi ðAi1 ; Ai2 ; . . . ; Aid Þ is a polynomial of degree bih in the coordinates Aih ; 1 6 h 6 d. Proof. The first partial derivatives of dðx; xi Þ, the distance between x ¼ ðt1 ; t2 ; . . . ; td Þ and xi ¼ ðt i1 ; t i2 ; . . . ; t id Þ, are

@dðx; xi Þ t h  t ih ¼  Aih ; @t h dðx; xi Þ

1 6 i 6 n;

1 6 h 6 d:

ð41Þ

Note that j Aih j6 1. The second partial derivatives are

@ 2 dðx; xi Þ @Aih dhk  Aik Aih ¼ ¼ ; @t k @t h @t k dðx; xi Þ

1 6 i 6 n;

1 6 h; k 6 d:

ð42Þ

The lemma is then trivial for bih ¼ 0; 1 6 h 6 d, i.e. j bi j¼ 0, and is true for j bi j¼ 1 and j bi j¼ 2 from Eqs. (41) and (42), respectively. We proceed by induction on j bi j. Without loss of generality, we assume j bi jP 3 and bi1 P 1; on the contrary, we could develop the argument for any bih P 1; h – 1. We have

h i h i 2jb j Dbi dðx; xi Þ ¼ D1;0;...;0 Dðbi1 1Þ;bi2 ;...;bid dðx; xi Þ ¼ D1;0;...;0 d i ðx; xi Þpðbi1 1;bi2 ;...;bid Þ ðAi1 ; Ai2 ; . . . ; Aid Þ 1jbi j

¼ ð2 j bi jÞd

2jbi j

ðx; xi ÞAi1 pðbi1 1;bi2 ;...;bid Þ ðAi1 ; Ai2 ; . . . ; Aid Þ þ d

ðx; xi Þ

d X @pðbi1 1;bi2 ;...;bid Þ ðAi1 ; Ai2 ; . . . ; Aid Þ @Aih : @Aih @t 1 h¼1

ð43Þ The derivatives of Aih with respect to t 1 are given by (42); hence (43) becomes

"

1jbi j

bi

D dðx; xi Þ ¼ d

1jbi j

¼d

d X @pðbi1 1;bi2 ;...;bid Þ ðAi1 ; Ai2 ; . . . ; Aid Þ ðx; xi Þ ð2 j bi jÞAi1 pðbi1 1;bi2 ;...;bid Þ ðAi1 ; Ai2 ; . . . ; Aid Þ þ ðdh1  Ai1 Aih Þ @Aih h¼1

#

ðx; xi Þpðbi1 ;bi2 ;...;bid Þ ðAi1 ; Ai2 ; . . . ; Aid Þ; ð44Þ

which completes the proof. h Lemma 8.2. For j bi j< l; bi

l

ljbi j

D d ðx; xi Þ ¼ d

l > 1,

ðx; xi Þpbi ðAi1 ; Ai2 ; . . . ; Aid Þ:

ð45Þ

Proof. The first partial derivatives are

" #l=2 l d d i X @d ðx; xi Þ @ X l l2 @ h l2 2 ¼ ðt k  t ik Þ ¼ d ðx; xi Þ ðt k  t ik Þ2 ¼ lðth  t ih Þd ðx; xi Þ: @th @th k¼1 @t 2 h k¼1

ð46Þ

These derivatives exist and are continuous for all x 2 X with the exception of x ¼ xi . The existence and the values of the derivatives at xi depend on l. In fact, considering the left and right limits of the derivatives along the hth coordinate axis, they are infinite for l < 1, whereas they do not coincide for l ¼ 1. On the contrary, for l > 1 the limit exists and equals zero. Conl sidering derivatives of higher order of d ðx; xi Þ; l > 1, we look for a result similar to (40). We then proceed by induction on j bi j. The conclusion is trivial for bih ¼ 0; 1 6 h 6 d, i.e. j bi j¼ 0, and is true for j bi j¼ 1 because from (46) l

@d ðx; xi Þ ðt h  t ih Þ l1 l1 ¼ ld ðx; xi Þ ¼ ld ðx; xi ÞAih : @th dðx; xi Þ Without loss of generality, assume that bi1 P 1. We have by (47)

h i h i l l l1 Dbi d ðx; xi Þ ¼ Dðbi1 1Þ;bi2 ;...;bid D1;0;...;0 d ðx; xi Þ ¼ lDðbi1 1Þ;bi2 ;...;bid d ðx; xi ÞAi1 ¼l

bX i1 1

bi2 X

m1 ¼0 m2 ¼0





bid X bi2 b bi1  1 l1    id  Dm1 ;m2 ;...;md d ðx; xi ÞDðbi1 1m1 Þ;ðbi2 m2 Þ...;ðbid md Þ Ai1 : m m m 1 2 d m ¼0 d

ð47Þ

9260

G. Allasia, C. Bracco / Applied Mathematics and Computation 218 (2012) 9248–9260

By the induction hypothesis, we have l1

Dm1 ;m2 ;...;md d

l1ðm1 þm2 þþmd Þ

ðx; xi Þ ¼ d

ðx; xi Þ  pðm1 ;m2 ;...;md Þ ðAi1 ; Ai2 ; . . . ; Aid Þ:

ð48Þ

Moreover, applying the result in (40) gives

Dðbi1 1m1 Þ;ðbi2 m2 Þ;...;ðbid md Þ Ai1 ¼ Dðbi1 m1 Þ;ðbi2 m2 Þ;...;ðbid md Þ dðx; xi Þ 1ðbi1 m1 Þðbi2 m2 Þðbid md Þ

¼d

dðx; xi Þ  pðbi1 m1 ;bi2 m2 ;...;bid md Þ ðAi1 ; Ai2 ; . . . ; Aid Þ:

ð49Þ

From (48) and (49) it follows l

Dbi d ðx; xi Þ ¼ l

bP bi2 i1 1 P



m1 ¼0m2 ¼0

bid P



md ¼0

bi1  1 m1





bi2 b    id m2 md

h l1ðm1 þm2 þþmd Þ  d ðx; xi Þpðm1 ;m2 ;...;md Þ ðAi1 ; Ai2 ; . . . ; Aid Þ 1ðbi1 m1 Þðbi2 m2 Þðbid md Þ

d

ljbi j

¼d

ðx; xi Þ pðbi1 m1 ;bi2 m2 ;...;bid md Þ ðAi1 ; Ai2 ; . . . ; Aid Þ

i

ðx; xi Þpðbi1 ;bi2 ;...;bid Þ ðAi1 ; Ai2 ; . . . ; Aid Þ;

which is the result in (45). h l

Proof of Theorem 5.1. From (45) we deduce that the derivatives of d ðx; xi Þ with order less than l are continuous and ljbi j bounded on X. In fact, the factor d is continuous on X and the polynomial pðbi1 ;bi2 ;...;bid Þ ðAi1 ; Ai2 ; . . . ; Aid Þ is continuous on X  fxi g. Moreover, we can assume Dbi dl ðxi ; xi Þ ¼ 0, because pðbi1 ;bi2 ;...;bid Þ ðAi1 ; Ai2 ; . . . ; Aid Þ is bounded at xi being ljbi j l j Aih j6 1; 0 6 h 6 d, and d ðx; xi Þ goes to zero as x ! xi . To conclude we observe that the function aðx; yÞ ¼ d ðx; yÞ satisfies the requirements of Theorem 4.1 with q ¼ dle if l is not an integer or l  1 if l is an integer. h References [1] G. Allasia, A class of interpolating positive linear operators: theoretical and computational aspects, in: S.P. Singh (Ed.), Approximation Theory, Wavelets and Applications, Kluwer, Dordrecht, 1995, pp. 1–36. [2] G. Allasia, Cardinal basis interpolation on multivariate scattered data, Nonlinear Anal. Forum 6 (1) (2001) 1–13. [3] G. Allasia, Simultaneous interpolation and approximation by a class of multivariate positive operators, Numer. Algor. 34 (2003) 147–158. [4] G. Allasia, C. Bracco, Hermite and Birkhoff interpolation to multivariate scattered data, Quaderno Scientifico del Dipartimento di Matematica, University of Turin, vol. 25, 2008, pp. 1–23 [5] R.E. Barnhill, R.P. Dube, F.F. Little, Properties of Shepard’s surfaces, Rocky MT. J. Math. 13 (2) (1983) 365–382. [6] I.S. Berezin, N.P. Zhidkov, Computing Methods, two vols., Pergamon Press, Oxford, 1965; Translation of Metody vychislenii, Fizmatgiz, Moscow, 1959. [7] B.D. Bojanov, H.A. Hakopian, A.A. Sahakian, Spline Functions and Multivariate Interpolation, Kluwer, Dordrecht, 1993. [8] E. Cheney, Multivariate approximation theory: selected topics, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 51, SIAM, Philadelphia, 1986. [9] P.J. Davis, Interpolation and Approximation, Dover, New York, 1975. [10] G.E. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific Publishers, Singapore, 2007. [11] R. Franke, G. Nielson, Smooth interpolation of large sets of scattered data, Numer. Methods Eng. 11 (15) (1980) 1691–1704. [12] R. Franke, Scattered data interpolation: tests of some methods, Math. Comput. 33 (1982) 181–200. [13] W.J. Gordon, J.A. Wixom, Shepard method of ‘‘’metric interpolation’’ to bivariate and multivariate interpolation, Math. Comput. 32 (141) (1978) 253– 264. [14] A. Iske, Reconstruction of functions from generalized Hermite–Birkhoff data, in: C.K. Chui, L.L. Schumaker (Eds.), Approximation Theory VIII, Vol. 1: Approximation and Interpolation, World Scientific Publishing, Singapore, 1995, pp. 257–264. [15] R.A. Lorentz, Multivariate Birkhoff interpolation, Springer, Heidelberg, 1992. [16] R.A. Lorentz, Multivariate Hermite interpolation by algebraic polynomials: a survey, J. Comput. Appl. Math. 122 (2000) 167–201. [17] R.J. Renka, Multivariate interpolation of large sets of scattered data, ACM Trans. Math. Softw. 2 (14) (1988) 139–148. [18] L.L. Schumaker, Fitting surfaces to scattered data, in: G.G. Lorentz, C.K. Chui, L.L. Schumaker (Eds.), Approximation Theory II, Academic Press, New York, 1976, pp. 203–268. [19] D. Shepard, A two-dimensional interpolation function for irregularly spaced data, in: Proceedings 23rd Nature Conference, ACM, Brandon/Systems Press Inc., Princeton, 1968, pp. 517–524. [20] H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2005. [21] Z.M. Wu, Hermite–Birkhoff interpolation of scattered data by radial basis functions, Approx. Theory Appl. 8 (2) (1992) 1–10.