As in the case of positive semi-de¬nite functions, conditionally positive semi-de¬nite

functions can be characterized to be integrally conditionally positive semi-de¬nite. See

Proposition 6.4 for the corresponding result on positive semi-de¬nite functions.

Proposition 8.6 Let be continuous. Then is conditionally positive semi-de¬nite of

order m if and only if

(x ’ y)γ (x)γ (y)d xd y ≥ 0 (8.3)

Rd Rd

100 Conditionally positive de¬nite functions

∞

for all γ ∈ C0 (Rd ) that satisfy

γ (x) p(x)d x = 0 p ∈ πm’1 (Rd ).

for all (8.4)

Rd

Proof Suppose possesses the stated property. If we choose a nonnegative even function

∞

g ∈ C0 (Rd ) with g L 1 (Rd ) = 1 and set g (x) = d g( x) then we know that

f (x) = lim f (y)g (x ’ y)dy

’∞ Rd

N

for every continuous f . If x1 , . . . , x N ∈ Rd and ± ∈ C N such that ± j p(x j ) = 0 for

j=1

all p ∈ πm’1 (Rd ) are given then the functions

N

γ (x) := ± j g (x ’ x j )

j=1

∞

are in C0 (Rd ) and satisfy

N

γ (x) p(x)d x = ± j p(x + x j )g (x)d x = 0

Rd Rd j=1

for all p ∈ πm’1 (Rd ). Thus we ¬nd that

0¤ (x ’ y)γ (x)γ (y)d xd y

Rd Rd

N

= ± j ±k (x ’ y ’ (x j ’ xk ))g (x)g (y)d xd y,

Rd Rd

j,k=1

which converges for ’ ∞ to

N

± j ±k (x j ’ xk ).

j,k=1

Conversely, let us suppose that is conditionally positive semi-de¬nite of order m. We

want to employ Riemann sums to show that (8.3) is satis¬ed for all γ ∈ C0 (Rd ) with (8.4).

Unfortunately, if m > 0 then the discretizations gained by Riemann sums will not satisfy

(8.1) in general. Hence, we have to modify the approach. Fix a γ ∈ C0 (Rd ) that satis¬es

(8.4). Let K be a compact cube that contains the support of γ . Then we have to discretize

the integral

γ (x)γ (y) (x ’ y)d xd y. (8.5)

K K

A scaling argument shows that without loss of generality we can assume K to be the unit

cube [0, 1]d .

We divide K into N equally sized subcubes of volume 1/N and pick an x (N ) from each

j

of these subcubes. Let Q be the dimension of πm’1 (Rd ) and p1 , . . . , p Q be a basis for

πm’1 (Rd ). Let N ≥ Q. We choose the points {x (N ) } and their ordering in such a way that

j

8.1 De¬nition and basic properties 101

the ¬rst Q points are always the same for every N , i.e. x1 ) = x1 , . . . , x Q ) = x Q for all

(N (N

N ≥ Q. Moreover, these points should be chosen to be πm’1 (Rd )-unisolvent. Then we

discretize the integrals (8.4) to de¬ne the numbers

N

1

A(N ) := γ (x (N ) ) pk (x (N ) ), 1¤k¤Q N ≥ Q.

and

k j j

N

j=1

From the conditions imposed on γ we know that A(N ) tends to zero for N ’ ∞ and this

k

holds for all 1 ¤ k ¤ Q. Since {x1 , . . . , x Q } is πm’1 (Rd )-unisolvent, we ¬nd for every

N ≥ Q a unique vector β (N ) ∈ R Q with

Q

β (N ) pk (x j ) = A(N ) , 1 ¤ k ¤ Q.

j k

j=1

This β (N ) is obviously given by β (N ) = P ’1 A(N ) if P = ( pk (x j )). Hence each coef¬cient

β (N ) tends to zero as N ’ ∞. Finally let us de¬ne the corrected coef¬cients

j

§

⎪ 1 γ (x (N ) ) ’ β (N )

⎨ for 1 ¤ j ¤ Q,

j j

± j := N

⎪1

© γ (x (N ) ) for Q + 1 ¤ j ¤ N .

j

N

These coef¬cients satisfy condition (8.1) by construction:

Q

N N

1

± j pk (x (N ) ) γ (x (N ) ) pk (x (N ) ) ’ β (N ) pk (x (N ) )

=

j j j j j

N

j=1 j=1 j=1

A(N ) A(N )

= ’ =0

k k

for 1 ¤ k ¤ Q. Hence, we can insert them into the quadratic form for , obtaining

N N

± j ±k (x (N ) ’ xk ) )

(N

0¤ j

j=1 k=1

N N

1

γ (x (N ) )γ (xk ) ) (x (N ) ’ xk ) )

(N (N

= j j

N2

j=1 k=1

Q N

1

β (N ) γ (xk ) ) (x j ’ xk ) )

(N (N

’ j

N

j=1 k=1

Q

N

1

γ (x (N ) ) βk ) (x (N ) ’ xk )

(N

’ j j

N

j=1 k=1

Q

β (N ) βk ) (x j ’ xk ).

(N

+ j