±! Rd

|±|=

!

= D 2± f (x ’ y) f » (y)dy

±! Rd

|±|=

= f (x ’ y) f » (y)dy

Rd

N

= »j (y ’ x j ) f (x ’ y)dy

d,

Rd

j=1

N

= (’1) » j f (x ’ x j ),

j=1

proving the result in this case. For a general f ∈ BL (Rd ) we ¬x x ∈ Rd and choose a

compact set K ⊆ Rd such that x ’ x j ∈ K for 1 ¤ j ¤ N . For an arbitrary > 0 we

∞

choose a g ∈ C0 (Rd ) according to Theorem 10.40. Then two applications of the triangle

inequality show that the absolute value of the difference in the two sides of (10.13) can be

N

j=1 |» j | + | f » |BL (Rd ) , which tends to zero with ’ 0.

bounded by

This is the major step in proving several things. For example we can now readily conclude

that every function f from the Beppo Levi space is slowly increasing; we will derive a

representation formula for f and ¬nally we will show that such a function has to be in the

native space of the thin-plate splines.

=

To achieve all these goals we use a now familiar concept. Suppose that

{ξ1 , . . . , ξ Q } ⊆ R is π ’1 (R )-unisolvent and p1 , . . . , p Q is a Lagrange basis of π ’1 (R )

d d d

166 Native spaces

. Next we de¬ne for a ¬xed x ∈ Rd the functional

with respect to

Q

» := δ’x ’ pk (x)δ’ξk ,

k=1

which is very similar to the functional δ(x) employed earlier. To see that » annihilates

polynomials we can simply apply it to the basis qk := pk (’·). Denote by the projection

= π ’1 (Rd ) .

Theorem 10.42 Let > d/2. Every function f ∈ BL (Rd ) has the representation

f (x) = f (x) + ( f, G(·, x))BL (Rd ) , x ∈ Rd .

Here G is the function (10.4) for the basis functions speci¬ed in (10.11). This representation

means in particular that f is slowly increasing.

Proof The special choice of » made in the paragraph before this theorem together with

Theorem 10.41 shows that, for ω ∈ Rd ,

Q

f (ω + x) = p j (x) f (ω + ξ j ) + ( f, f » (ω ’ ·))BL (Rd ) (10.14)

j=1

with

Q

f » (ω) = (ω + x) ’ (ω + ξ j ).

p j (x)

d, d,

j=1

But the de¬nition of G, the fact that d, is even, and the fact that the generalized derivative

±

of d, coincides with the usual one outside zero allow us to conclude that D1 G(ω, x) =

(’1)|±| D ± f » (’ω). Hence setting ω = 0 in (10.14) gives the stated representation.

Finally, since the Beppo Levi semi-norm and the native space semi-norm coincide on the

native space we see that the Beppo Levi semi-norm of G(·, x) grows at most as a polynomial

in x and so does f .

Theorem 10.43 For > d/2 let d, = φd, ( · 2 ) be the thin-plate spline de¬ned in

(10.11). If d, is considered to be a conditionally positive de¬nite function of order

then the associated native space is the Beppo Levi space of order , i.e. N d, (Rd ) ≡

N d, ,π ’1 (Rd ) (Rd ) = BL (Rd ) and the semi-inner products are the same.

Proof We know already that the native space is contained in the Beppo Levi space. More-

over, since the semi-norms coincide on this subspace the native space is a complete subspace,

meaning that every Cauchy sequence has a (not necessarily unique) limit. This in turn means

that if the native space is not the whole Beppo Levi space then there must be an element

f ∈ BL (Rd ) that is orthogonal to the native space. The representation formula stated in

Theorem 10.42 now gives f ∈ π ’1 (Rd ).

10.6 An embedding theorem 167

10.6 An embedding theorem

Since in several cases the native space of a conditionally positive de¬nite kernel does not

coincide with a classical function space, it is important to know properties such as the

smoothness of the functions in the native space in advance, given only information about

the kernel itself. By construction we know already that N ( ) ⊆ C( ). Now we want to

see how the smoothness of the kernel is inherited by the native space.

To this end, we use the forward differences already introduced in (7.4), with a slightly

different notation,

k

k

f (r + j h),

(’1)k’ j

f (r ) :=

k,h

j

j=0

and their multivariate versions

x1 · · · xd f (x)

f (x) :=

±,h ±1 ,h ±d ,h

for ± = (±1 , . . . , ±d )T ∈ Nd and x = (x1 , . . . , xd )T ∈ Rd . Here x j means that acts

0

with respect to the x j -variable. From the property of the univariate forward difference, it

obviously follows that

lim h ’|±| f (x) = D ± (x)

±,h

h’0

if f is |±|-times continuously differentiable around x.

Lemma 10.44 Suppose that ⊆ Rd is open and that ∈ C 2k ( — ) is a conditionally

positive de¬nite symmetric kernel with respect to P ⊆ C k ( ). Then the function G(·, ·) from

(10.4) is k-times continuously differentiable with respect to the second argument, and for

± ±

every x ∈ and every ± ∈ Nd with |±| ¤ k the function D2 G(·, x) is in N ( ). Here D2

0

means that we differentiate with respect to the second argument.

Proof Obviously G possesses k continuous derivatives with respect to its second argument.

±

Moreover, D2 G(·, x) is in C k ( ) as a function of the ¬rst argument for all |±| ¤ k and all

x∈ .

Fix ± ∈ Nd with |±| ¤ k and x ∈ . De¬ne the function •n := ±,1/n,2 G(·, x). Here,

0

the additional 2 in the subscript means that acts with respect to the second argument of

G. Since G(·, x) ∈ F ( ) for every x ∈ , we also have •n ∈ F ( ). Hence we have a

y

representation of the form •n = »n (·, y) with »n ∈ L P ( ). The de¬nition also ensures

±

that •n (y) = »u (y, u) ’ D2 G(y, x). Now, •n is a Cauchy sequence in F ( ). Because

n

(•n , •m ) = »u »v (u, v) ’ D1 D2 G(x, x) =: c

±±

nm

for m, n ’ ∞, we have