bounded Lebesgue function. This is done by employing norming sets once again.

Lemma 17.11 Suppose that the knot set X = {x1 , . . . , x N } ⊆ S d’1 has ¬ll distance h X ¤

1/(2m). Then, Z = span{δx : x ∈ X } is a norming set for πm (S d’1 ) with norming constant

c = 1/2.

Proof The proof uses the same ideas as the proof of the corresponding result in Rd . Since

the restriction of any spherical polynomial Y ∈ πm (S d’1 ) to a great circle is a univariate

trigonometric polynomial of degree less than or equal to m, we can apply the classical

Bernstein inequality to get

|Y (x) ’ Y (y)| ¤ m dist(x, y) Y L ∞ (S d’1 ) , x, y ∈ S d’1 .

17.3 Error estimates 315

Hence, if Y ∈ πm (S d’1 ) satis¬es Y L ∞ (S d’1 ) = 1 then there exists a point x ∈ S d’1 such that

|Y (x)| = 1. By the condition imposed on X we can ¬nd a data site x j such that dist(x, x j ) ¤

h X ¤ 1/(2m). This, together with Bernstein™s inequality, shows that |Y (x) ’ Y (x j )| ¤ 1/2

or |Y (x j )| ≥ 1/2.

From the general theory on norming sets, in particular Theorem 3.4, we can immediately

conclude

Corollary 17.12 Suppose the knot set X = {x1 , . . . , x N } ⊆ S d’1 has ¬ll distance h X ¤

1/(2m). Then there exist functions u j : S d’1 ’ R such that

N

u j (x)Y (x j ) = Y (x) for all Y ∈ πm (S d’1 ) and x ∈ S d’1 ,

(1) j=1

N

|u j (x)| ¤ 2 for all x ∈ S d’1 .

(2) j=1

We will use this “global” (if compared to the Rd case) result to derive our ¬rst bound on

the interpolation error. We express the error in terms of the Fourier coef¬cients of the basis

kernel.

Theorem 17.13 Suppose that the kernel has only positive Fourier coef¬cients a ,k , which

satisfy the decay condition (17.4). Suppose further that X = {x1 , . . . , x N } ⊆ S d’1 has ¬ll

distance 1/(2m + 2) < h X ¤ 1/(2m). Then the error between f ∈ N (S d’1 ) and its inter-

polant s f,X can be bounded by

∞

9

| f (x) ’ s f,X (x)| ¤ N (S d’1 ) , x ∈ S d’1 .

2 2

a N (d, ) f

ωd’1 =m+1

Proof As usual, at the start we bound the interpolation error by | f (x) ’ s f,X (x)| ¤

P ,X (x) f N ( ) . Next, also as usual, we use the cardinal functions {u — } from Theorem

j

11.1 and the kernel expansion (17.2) to express the power function as

2

∞ N (d, ) N

u — (x)Y ,k (x j )

= Y ,k (x) ’ .

2

P ,X (x) a ,k j

=0 k=1 j=1

In the ¬nal step we employ the minimal property of the power function from Theorem 11.5

and replace the functions {u — } by the functions {u j } from Corollary 17.12. Furthermore, we

j

set u 0 (x) := ’1 and x0 = x, to derive

2

∞ N (d, ) N

¤ Y ,k (x) ’

2

P ,X (x) a u j (x)Y ,k (x j )

,k

=m+1 k=1 j=1

∞ N (d, ) N

¤ a u i (x)u j (x)Y ,k (xi )Y ,k (x j )

=m+1 k=1 i, j=0

∞ N

N (d, )

= P (d; xiT x j ).

a u i (x)u j (x)

ωd’1

=m+1 i, j=0

316 Interpolation on spheres and other manifolds

The last equality follows from Lemma 17.3. If ¬nally we take into account that the gener-

alized Legendre polynomial P is bounded by one and that

2

N N

|u i (x)u j (x)| ¤ 1 + |u j (x)| ¤ 9,

i, j=0 j=1

then by Corollary 17.12 we have completed the proof.

It is now easy to express the error estimates in terms of the ¬ll distance if additional

assumptions on the decay of the Fourier coef¬cients are made.

Corollary 17.14 Suppose that the assumptions of Theorem 17.13 hold.

(1) If a N (d, ) ¤ c(1 + )’± with ± > 1 then

(±’1)/2

f ’ s f,X ¤ Ch X N (S d’1 ) .

f

L ∞ (S d’1 )

(2) If a N (d, ) ¤ ce’±(1+ ) with ± > 0 then

¤ Ce’±/(4h X ) f

f ’ s f,X N (S d’1 ) .

L ∞ (S d’1 )

Proof In the ¬rst case, the assumption imposed on a gives

∞ ∞

c 2±’1 ±’1

c

(1 + )’± d = (1 + m)’±+1 =

a N (d, ) ¤ c h.

1’± ±’1 X

m

=m+1

In the second case, the same argument yields

∞ ∞

c ’±(1+m) c

e’±(1+ ) d = = e’±/(2h X ) .

a N (d, ) ¤ c e

± ±

m

=m+1

By now, it should be clear that other results, for example those on doubling the approxi-

mation order, can be carried over to the sphere in the same way. We leave the details to the

reader.

17.4 Interpolation on compact manifolds

The sphere is one possible example of a compact smooth manifold. In this short section we

will point out some ideas on how the results that we have reached so far can be extended

to other manifolds. As in the last two sections we will concentrate on positive de¬niteness

and error estimates.

For the convenience of the reader we review the necessary material on manifolds. A good

source for this is the book [30] by Boothby.

De¬nition 17.15 A set M ⊆ Rd is called a topological manifold of dimension n if it is a

Hausdorff space with a countable basis of open sets such that for every x ∈ M there exist an

open set U ⊆ M with x ∈ U and a mapping • : U ’ Rn that maps U homeomorphically

17.4 Interpolation on compact manifolds 317

to the open set V := •(U ) ⊆ Rn . The pair (U, •) is called a coordinate neighborhood of

x or a chart. A chart is of class C k if • ’1 ∈ C k (•(U )). A collection A = {(U± , •± )} of