General Methods for Approximation and Interpolation
3.1 Polynomial schemes
3.1 Polynomial schemes
The most frequently employed techniques for multivariate approximation, other than radial basis functions, are straight polynomial interpolation, and piecewise polynomial splines. We begin with polynomial interpolation. There are various, highly specific techniques for forming polynomial interpolants. Very special considerations are needed indeed because as long as is a finite generic set of data sites from an open set in more than one dimension, and if we are interpolating from a polynomial space independent of , there can always be singularity of the interpolation problem. That is, we can always find a finite set of sites that causes the interpolation problem to be singular, whenever the dimension is greater than one and the data sites can be varied within an open subset of the underlying Euclidean space.
This is a standard result in multivariate interpolation theory and it can be shown as follows. Suppose that is such that the interpolation matrix for a fixed polynomial basis, call the matrix A, is nonsingular. If stems from an open set in two or more dimensions, two of ’s points can be swapped, causing a sign change in det A, where for the purpose of the swap the two points can be moved along paths that do not intersect. Hence there must be a constellation of points for which det A vanishes, det A being continuous in each ξ ∈ due
to the independence of the polynomial basis of the points in . So we have proved the result that singularity can always occur (see Mairhuber, 1956). Of course, this proof works for all continuous finite bases, but polynomials are the prime example for this case.
As a consequence of this observation, we need either to impose special re- quirements on the placement of – which is nontrivial and normally not very attractive in applications – or to make the space of polynomials dependent on
, a more natural and better choice.
The easiest cases for multivariate polynomial interpolation with prescribed geometries of data points are the tensor-product approach (which is useless in most practical cases when the dimension is large because of the exponential increase of the required number of data and basis functions) and the interpolation e.g. on intersecting lines. Other approaches admittingm scattered data have been given by Goodman (1983), Kergin (1980), Cavaretta, Micchelli and Sharma (1980) and Hakopian (1982). All these approaches have in common that they yield unique polynomials inPm−1−ν
n , i.e. polynomials of total degreem−1−ν in
n unknowns, where ν < m varies according to the type of approach. They also have in common the use of ridge functions as a proof technique for establishing their properties, i.e. forming basis functions for the polynomial spaces which involve functions g(λ · x ) where λ and x are from Rn so that this function is
38 3. General methods
constant in directions orthogonal to λ and g ∈ C m−1−ν(R). The approach by
Goodman is the most general among these. The remarkable property of Kergin interpolation is that it simplifies to the standard Hermite, Lagrange or Taylor polynomials in one dimension, as the case may be. The work by Cavarettaet al.
is particularly concerned with the question which types of Hermite data (i.e. data involving function evaluations and derivatives of varying degrees) may be generalised in this way.
A completely different approach for polynomial interpolation in several un- knowns is due to Sauer and Xu (1995) who use divided differences represented in terms of simplex splines and directional derivatives to express the polyno- mials. Computational aspects are treated in Sauer (1995), see also the survey paper, Gasca and Sauer (2000).
The representations of the approximants are usually ill-conditioned and there- fore not too useful in practical applications. Some convergence results for the approximation method are available in the literature (Bloom, 1981, Goodman and Sharma, 1984).
The interpolation of points on spheres by polynomials has been studied by Reimer (1990) including some important results about the interpolation oper- ator. The key issue is here to place the points at which we interpolate suitably on the sphere. ‘Suitably’ means on one hand that the interpolation problem is well-posed (uniquely solvable) and on the other hand that the norm of the interpolation operator does not grow too fast with increasing numbers of data points. The former problem is more easily dealt with than the latter. It is eas- ier to distribute the points so that the determinant of the interpolation matrix is maximised than to find point sets that give low bounds on operator norms. Surprisingly, the points that keep the operator norms small do not seem to be distributed very regularly, while we get a fairly uniform distribution if for instance the potentials in the three-dimensional setting
ξ =ζ
1
ξ − ζ
are minimised with a suitable norm on the sphere. This work is so far only avail- able computationally for the two-dimensional sphere in R3, whereas theoretic
analysis extends beyond n = 3.
Another new idea is that of de Boor and Ron which represents the interpo- lating polynomial spaces and is dependent on the given data points. In order to explain the various notions involved with this idea, we need to introduce some simple and useful new notations now. They include the so-called least term of an analytic function – it is usually an exponential – and the minimal totality of a set of functionals, which is related to our interpolation problem.
3.1 Polynomial schemes 39
Definition 3.1. We call the least term f ↓ of a function f that is analytic at
zero the homogeneous polynomial f ↓ of largest degree j such that
f ( x ) = f ↓( x ) + O( x j +1), x → 0.
Also, for any finite-dimensional space H of sufficiently smooth functions, the least of the space H is
H ↓ := { f ↓ | f ∈ H }.
This is a space of polynomials.
Let P∗ be a space of linear functionals on the continuous functions. We recall that such a space P∗ of linear functionals is ‘minimally total’ for H if for any
h ∈ H , λh = 0 ∀λ ∈ P∗ implies h = 0, and if, additionally, P∗ is the smallest such space. Using Definition 3.1 and the notion of minimal totality, de Boor and Ron (1990) prove the following important minimality property of H ↓. Here,
the overline means, as is usual, complex conjugation.
Proposition 3.1. Among all spaces P of polynomials defined on Cn which
have the property that P∗ is minimally total for H, the least H ↓ is one of least
total degree, that is, contains the polynomials of smallest degree.
The reason why Proposition 3.1 helps us to find a suitable polynomial space for interpolation when the set of data is given is that we can reformulate the interpolation problem, which we wish to be nonsingular, in a more suitable form. That is, given that we wish to find a polynomial q from a polynomial space Q, say, so that function values on are met, we can represent the interpolation conditions alternatively in an inner product form as the requirement
f ξ = q∗ exp
ξ · (· )
, ξ ∈ .Here, the first · in the exponential’s argument denotes the standard Euclidean inner product, while the · in parentheses denotes the argument to which the functional q∗ is applied. The latter is, in turn, defined by application to any sufficiently smooth p through the formula
q∗ p =
α∈Zn + 1 α!
Dα q
(0) ·
Dα p
(0),using standard multiindex notation for partial derivatives
Dα =
∂ α1 ∂ x α1 1 , ∂ α2 ∂ x α2 2 , . . . , ∂ αn ∂ x αn n
40 3. General methods
and α! = α1! · α2!. . . αn!, a notation that occurs often in the book. This func-
tional is well-defined, whenever p is a function that is sufficiently smooth at the origin. It implies that our polynomial interpolation problem, as prescribed through Q and , is well-posed (i.e. uniquely solvable) if and only if the dual problem of interpolation from H :=
ξ ∈ aξ exp
ξ · (· )
aξ ∈ C
withinterpolation conditions defined through q∗ is well-posed. Hence the minimal totality of the set H ↓ can be used to prove the following important result.
Theorem 3.2. Given a finite set of data ⊂ Cn , let H be as above. Then H ↓
is a polynomial space of least degree that admits unique interpolation to data defined on .
The authors de Boor and Ron state this result more generally for Hermite interpolation, i.e. it involves interpolation of derivatives of various degrees and various centres.
There is also an algorithm for computing the least of a space that is a re- cursive method and is closely related to the Gram–Schmidt orthogonalisation procedure. We refer to the paper by de Boor and Ron for the details of this algorithm.
We give a few examples for the polynomials involved in two dimensions. If
contains just one element, then
H =
aξ exp
ξ · (·)
,with
exp
ξ · (·)
= 1 + ξ · (·) + 12
ξ · (·)
2 + · · · .Thus H ↓ = span{1}. Therefore our sought polynomial space, call itP, is P02, i.e.
constant polynomials in two variables. In general we let Pk
n be all polynomials
in Pn of total degree at most k . If contains two elements ξ and τ , then
H = {aξ exp(ξ · (·)) + aτ exp(τ · (·))},
hence H ↓ = span{1,(·) · (ξ − τ )}. Therefore P = P11 ◦ (λ·), where ◦ denotes
composition and where the vector λ is parallel to the affine hull of , a one-
dimensional object. If || = 3, thenP = P2
1◦(λ·) orP 1
2, depending on whether
the convex hull of is a line parallel to the vectorλor not. Finally, if contains
four elements and they are on a line, P = P3
1 ◦ (λ·); otherwise P 1
2 ⊂ P ⊂ P 2 2.
E.g. if = {0, ξ , τ , ξ + τ }, ξ = first coordinate unit vector, τ = second coordinate unit vector, then P is the space of bi-linear polynomials, i.e. we have