for all c ∈ Rd and all x ∈ .

306 Generalized interpolation

by a constant C > 0, the maximum principle

If all the functions a j,k are bounded on

for elliptic operators gives

C

u ’ su, ¤ u ’ su, + Lu ’ Lsu, L∞( ).

L∞( ) L ∞ (‚ )

±

Hence we have the following corollary to Theorem 16.15:

Corollary 16.17 If in addition to the assumptions of Theorem 16.15 the operator L is of

second order and elliptic and if

h = max{h X 1 , , h X 2 ,‚ }

then

u ’ su, ¤ Ch k’2 u N()

L∞( )

for all suf¬ciently small h.

16.4 Notes and comments

In this chapter we have aimed to demonstrate the potential of (generalized) scattered data

approximation. The ultimate goal is the solution of time-dependent partial differential equa-

tions with moving boundaries, where classical methods such as ¬nite elements encounter

severe problems because of the necessary remeshing. As already mentioned in Chapter 1,

¬rst promising steps can be found in Lorentz et al. [109] and Behrens and Iske [21]. Other

possible applications come from the ¬nancial sciences, where differential equations in

high-dimensional spaces have to be solved.

The crucial point in the ¬rst section was showing that the use of positive de¬nite kernels is

so ¬‚exible that it essentially makes no difference whether pure interpolation or more general

functionals are investigated. The Hermite“Birkhoff interpolation served as an example here.

It was initially investigated by Wu [202] and Narcowich and Ward [146].

Besides the collocation method introduced here, which obviously produces symmetric

coef¬cient matrices and which was introduced by Fasshauer [56] and investigated by Franke

and Schaback [61,62], there is another method on the market that is often used. This method

was introduced by Kansa [96,97] in 1990 and simply uses the Ansatz from pure interpolation,

N

± j (·, x j ),

s=

j=1

but determines the coef¬cients via general functionals » j (s) = » j ( f ), 1 ¤ j ¤ N . Unless

the functionals » j are point evaluation functionals at the sites x j , this method produces a

nonsymmetric coef¬cient matrix, which might even become singular; see Hon and Schaback

[85]. But these cases seem to be rare, and in all other cases Kansa™s method often behaves

better than the symmetric one.

16.4 Notes and comments 307

Collocation is not the only method that has been investigated for solving partial dif-

ferential equations, however. The present author studied in [193, 194] Galerkin methods.

In applications, the so-called dual reciprocity method combined with a boundary-element

method has often been used. The dual reciprocity method was introduced by Nardini and

Brebbia [150] in 1982 and brought into the context of radial basis functions by Golberg [71].

The idea behind this method is to homogenize the differential equation, for example by radial

basis functions, and then to use speci¬c boundary-element methods to solve the remaining

problem.

17

Interpolation on spheres and other manifolds

So far w have been concerned with interpolation on an arbitrary domain ⊆ Rd . However,

we have not used any topological information about . Instead, we have employed only

the fact that it is a subset of Rd . As a matter of fact, without having more information on

this is the only way. But many applications provide us with additional information on the

underlying domain. For example, problems coming from geology often relate to the entire

earth, so that the unit sphere would be an appropriate model and the additional information

should lead to a better approximant.

Hence, in this chapter, we want to give an introduction to the theory of scattered data

interpolation on spheres and other compact manifolds by radial or zonal functions.

17.1 Spherical harmonics

Generally, functions on the sphere are expressed as Fourier series with respect to an or-

thonormal family called spherical harmonics. In this section we will review the results on

these functions. Since this material is only necessary for the present chapter we did not in-

corporate it into Chapter 5. Moreover, we have to skip the proofs once again. The interested

reader is referred to M¨ ller™s book [140].

u

The domain of interest is the d-variate unit sphere S d’1 := {x ∈ Rd : x 2 = 1} ⊆ Rd .

It has surface area

2π d/2

ωd’1 = .

(d/2)

On S d’1 , we will employ the usual inner product

( f, g) L 2 (S d’1 ) := f (x)g(x)d S(x), (17.1)

S d’1

where d S(x) is given by the standard measure on the sphere.

The distance between two points x, y ∈ S d’1 is the geodesic distance, which is the

length of the shorter part of the great circle joining x and y or, in other words, dist(x, y) =

arccos(x T y).

308

17.1 Spherical harmonics 309

There are different ways of introducing spherical harmonics. We start by de¬ning the

set of spherical polynomials of degree , π (S d’1 ), as the restriction of the classical

d-variate polynomials of degree to the sphere, i.e. π (S d’1 ) := π (Rd )|S d’1 . Note that

a basis of π (Rd ) is no longer a basis for π (S d’1 ) because of the additional requirement

x 2 = x2 + · · · + x2 = 1. A possible basis for π (S d’1 ) is given by the set of spherical

d

2 1

harmonics.

De¬nition 17.1 The orthogonal complement of π ’1 (S d’1 ) in π (S d’1 ) with respect to the

inner product (17.1) is denoted by π ⊥ (S d’1 ). The spherical harmonics {Y ,k : 1 ¤ k ¤

’1

N (d, )} are an orthonormal basis for π ⊥ (S d’1 ).

’1

⊥

Here, we use π’1 (S d’1 ) = {0}, so that π’1 (S d’1 ) is the one-dimensional space of constants.

It is known that

if = 0,

1

2 +d ’2 +d ’3

π ⊥ (S d’1 )

N (d, ) = dim = if ≥ 1,

’1

’1

and that N (d, ) = O( d’2 ) for ’ ∞. Since π (S d’1 ) is the disjoint union of the

π ⊥ (S d’1 ) for 0 ¤ j ¤ , the dimension of the space π (S d’1 ) is given by

j’1

dim π (S d’1 ) = N (d, j) = N (d + 1, ),

j=0

which is easily established by induction on .

Another way of introducing spherical harmonics explains the name better. Remember

that a harmonic function f satis¬es f = 0. A spherical harmonic of order is in this

de¬nition the restriction of a homogeneous harmonic polynomial of degree to the sphere.

Since the spherical harmonics form an orthonormal basis for L 2 (S d’1 ), every function

f ∈ L 2 (S d’1 ) has a Fourier representation of the form

∞ N (d, )

f= = ( f, Y ,k ) L 2 (S d’1 ) ,

f ,k Y ,k with f ,k