k

0 m=0

Thus, by Lemma 7.9 we get

1

μn = lim t n d±k .

k’∞ 0

Moreover, the ±k have ¬nite total mass and this is also uniformly bounded:

k

1

±k = d±k = »k,m = μ0 .

0 m=0

Hence, Helly™s theorem (Theorem 5.34) now guarantees the existence of a ¬nite nonnegative

measure ± with total mass ± ¤ μ0 and a sequence ±k j , so that

1 1

f (t)d±k j =

lim f (t)d±

j’∞ 0 0

for all f ∈ C[0, 1]. Setting f (t) = t n ¬nishes the proof.

After these preparatory steps we are able to prove a characterization that was indepen-

dently treated by Bernstein in 1914 and 1928, by Hausdorff in 1921, and by Widder in

1931.

Theorem 7.11 (Hausdorff“Bernstein“Widder) A function φ : [0, ∞) ’ R is com-

pletely monotone on [0, ∞) if and only if it is the Laplace transform of a nonnegative

¬nite Borel measure ν, i.e. it is of the form

∞

e’r t dν(t).

φ(r ) = Lν(r ) =

0

92 Completely monotone functions

Proof In the introductory part of this chapter we have seen already that every φ that is

the Laplace transform of a nonnegative and ¬nite measure ν satis¬es (’1)k φ (k) (r ) ≥ 0 for

r > 0. Moreover, because ν is ¬nite, φ is also continuous at zero.

Conversely, let us assume that φ is completely monotone. For a ¬xed N ∈ N we consider

the sequences μn = φ(n/N ), n ∈ N0 . Since

m

μm = 1/N φ ,

k k

N

we ¬nd that

k m

»k,m = 1/N φ ≥0

k’m

(’1)k’m

m N

for 0 ¤ m ¤ k and k ∈ N0 , by Corollary 7.5. Proposition 7.10 gives for every N ∈ N a ¬nite

nonnegative Borel measure ± N on [0, 1] such that

1

n

φ = t n d± N , n ∈ N0 .

N 0

If we de¬ne the measurable maps TN : [0, 1] ’ [0, 1], t ’ t N , and S : [0, ∞) ’ (0, 1],

t ’ e’t , we can conclude, on the one hand, that

1 1

φ(n) = d± N =

nN

t n dTN (± N ).

t

0 0

Since both measures TN (± N ) and ±1 are ¬nite, we can use the approximation theorem of

Weierstrass to derive

1 1

f (t)d±1 = f (t)dTN (± N )

0 0

for all f ∈ C[0, 1], which gives by the uniqueness theorem T (± N ) = ±1 . On the other hand,

we have

∞ ∞

1 1

n

e’nt/N d S ’1 ±1 =: e’nt/N dν.

φ = t n d± N = t n/N d±1 =

N 0 0+ 0 0

Since ν = S ’1 ±1 inherits the properties of ±1 , it is nonnegative and ¬nite. Using that φ is

continuous and Lebesgue™s convergence theorem leads ¬nally to

∞

e’r t dν(t)

φ(r ) =

0

for all r ≥ 0.

For later reasons we now state the corresponding result for completely monotone func-

tions on (0, ∞). This will play an important role in the theory of conditionally positive

de¬nite functions. The difference from Theorem 7.11 is that the measure does not need to

be ¬nite. Actually, it is ¬nite if and only if φ is continuous at zero.

7.3 Schoenberg™s characterization 93

Corollary 7.12 A function φ is completely monotone on (0, ∞) if and only if there exists a

nonnegative Borel measure ν on [0, ∞) such that

∞

e’r t dν(t)

φ(r ) =

0

for all r > 0.

Proof The proof of the suf¬cient part follows as before by successive differentiation, which

can be justi¬ed because

t n e’r t = t n e’r t/2 e’r t/2 ¤ Cn,r e’r t/2

for all t ≥ 0 and Cn,r is uniformly bounded for all r in a ¬xed compact subset of (0, ∞).

For the necessary part, note that for each δ > 0 the function φ(· + δ) is completely

monotone on [0, ∞). Thus by Theorem 7.11 we ¬nd a measure ±δ such that

∞

e’r t d±δ (t).

φ(r + δ) =

0

eδt d±δ for A ∈ B([0, ∞)) then we can derive

If we de¬ne the measure ν by ν(A) = A

∞

e’r t dν(t)

φ(r ) = (7.5)

0

for r > δ. But from the uniqueness property of the Laplace transform we can conclude that

ν actually does not depend on δ > 0, so that (7.5) remains valid for r > 0.

7.3 Schoenberg™s characterization

After having established that completely monotone functions are nothing other than Laplace

transforms of nonnegative and ¬nite Borel measures, we turn to the connection between

positive semi-de¬nite radial and completely monotone functions, which was ¬rst pointed

out by Schoenberg in 1938.

Theorem 7.13 (Schoenberg) A function φ is completely monotone on [0, ∞) if and only

if := φ( · 2 ) is positive semi-de¬nite on every Rd .

2

The univariate function φ acts here as a d-variate function via φ( · 2 ), which differs

2

from our de¬nition of a radial function. We will reformulate the result from the point of

view of a positive de¬nite function after the proof.

Proof If φ is completely monotone on [0, ∞) then it has a representation

∞

e’r t dν(t)