0

94 Completely monotone functions

for some nonnegative ¬nite Borel measure ν, by Theorem 7.11. Thus the d-variate function

can be represented by

∞

e’

2

(x) = φ( x 2 ) = x 2t dν(t),

2

0

and is hence positive semi-de¬nite, as we have already seen in the introductory part of this

chapter.

Next, let us suppose that φ( · 2 ) is positive semi-de¬nite on every Rd . Since φ is

2

obviously continuous in zero, we know from Theorem 7.4 that it suf¬ces to show that

(’1)k k φ(r ) ≥ 0 for all k ∈ N0 and all r, h > 0. This can be done by induction on k. For

√

h

k = 0 we have to show that φ(r ) ≥ 0 for all r ∈ (0, ∞). To this end we choose x j = r/2 e j ,

1 ¤ j ¤ N , where e j denotes the jth unit coordinate vector in R N . Since φ( · 2 ) is positive

2

semi-de¬nite on every R we get

N

N

0¤ φ( x j ’ x = N φ(0) + N (N ’ 1)φ(r ),

2

2)

j, =1

because our special choice of data sites gives x j ’ x 2 = r for j = . Dividing by

2

N (N ’ 1) and letting N tend to in¬nity allows us to conclude that φ(r ) ≥ 0.

For the induction step it obviously suf¬ces to show that ’ 1 φ( · 2 ) is also positive

h 2

semi-de¬nite on every Rd , if φ( · 2 ) is positive semi-de¬nite on every Rd . To do this,

2

suppose that x1 , . . . , x N ∈ Rd and ± ∈ R N are given. We take the x j as elements of Rd+1

and de¬ne

xj, 1 ¤ j ¤ N,

√

y j :=

x j’N + hed+1 , N < j ¤ 2N ,

and

±j, 1 ¤ j ¤ N,

β j :=

’± j’N , N < j ¤ 2N .

Since φ( · is also positive semi-de¬nite on Rd+1 we have

2

2)

2N

0¤ β j βk φ( y j ’ yk 2 )

2

j,k=1

N N 2N

= ± j ±k φ( x j ’ ’ ± j ±k’N φ( x j ’ xk’N + h)

xk 2 ) 2

2 2

j=1 k=N +1

j,k=1

2N N

’ ± j’N ±k φ( x j’N ’ xk + h)

2

2

j=N +1 k=1

2N

+ ± j’N ±k’N φ( x j’N ’ xk’N 2

2)

j,k=N +1

7.3 Schoenberg™s characterization 95

N

=2 ± j ±k φ( x j ’ xk 2 ) ’ φ( x j ’ xk + h)

2

2 2

j,k=1

N

= ’2 ± j ±k h φ( x j ’ xk 2 ).

1

2

j,k=1

Thus ’ h φ( ·

1 2

2) is positive semi-de¬nite.

Again, we are more interested in positive de¬nite functions than in positive semi-de¬nite

ones. This time, a complete characterization of radial functions φ as being positive de¬nite

on every Rd is simpler than in the case of general positive de¬nite functions on a ¬xed Rd .

Theorem 7.14 For a function φ : [0, ∞) ’ R the following three properties are equiva-

lent:

(1) φ is positive de¬nite on every Rd ;

√

(2) φ( ·) is completely monotone on [0, ∞) and not constant;

(3) there exists a ¬nite nonnegative Borel measure ν on [0, ∞) that is not concentrated at zero, such

that

∞

e’r t dν(t).

2

φ(r ) =

0

Proof From Theorems 7.13 and 7.11 we know already that φ is positive semi-de¬nite on

√

every Rd if and only if φ( ·) is completely monotone and if and only if it has the stated

integral representation. Hence it only remains to discuss the additional properties. For

pairwise distinct x1 , . . . , x N ∈ Rd and ± ∈ R N \ {0} the quadratic form can be represented

by (7.2). Since the Gaussian is positive de¬nite we see that the function φ is not positive

de¬nite (and thus only positive semi-de¬nite) if and only if the measure ν is up to a constant

nonnegative factor the Dirac measure centered at zero. This shows the equivalence of the ¬rst

and the third property. Finally, the second and the third property are obviously equivalent.

We ¬nish this chapter by providing a further example, returning to inverse multiquadrics.

We mentioned earlier that the restriction β > d/2 in Theorem 6.13 is arti¬cial. The last

result allows us to get rid of this restriction. Moreover, the proof of positive de¬niteness

becomes much simpler than before.

Theorem 7.15 The inverse multiquadrics φ(r ) := (c2 + r 2 )’β are positive de¬nite func-

tions on every Rd provided that β > 0 and c > 0. Moreover, φ has the representation

∞

e’r t dν(t),

2

φ(r ) =

0

with measure ν allowing the representation

1 β’1 ’c2 t

dν(t) = t e dt.

(β)

96 Completely monotone functions

√

Proof Set f (r ) = φ( r ). Then f is completely monotone since

(’1) f ( ) (r ) = (’1)2 β(β + 1) · · · (β + ’ 1)(r + c2 )’β’ ≥ 0.

Since f is not constant φ must be positive de¬nite. For the representation, see the proof of

Theorem 6.13.

7.4 Notes and comments

Completely monotone functions are obviously closely related to absolutely monotone func-

tions, which satisfy f ( ) ≥ 0 for all . The latter were introduced by Bernstein [25] in 1914

and characterized as Laplace“Stieltjes integrals in 1928 [26]. Somewhat earlier than the