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Additional resources for Bernstein functions : theory and applications
F . C a/ D f . C a/ C . 1/a f . /: It follows directly from the deﬁnition of the set C that f . C a/ and a f are again in Cb . Thus, if f 2 K is extremal, both functions must be multiples of f . In particular, f . a/f . /; 0; a > 0: Letting ! a/. 0C/ D 1 and since f is continuous, all solutions of the functional equation f . a/f . / are of the form f . / D e t . / D e t where t 2 Œ0; 1/. This means that the extreme points of K are contained in the set ¹e t W t 0º. Since K is the closed convex hull of its extreme points, we conclude that there exists a measure supported in Œ0; 1/ such that Œ0; 1/ D 1 and Z f.
D e t . / D e t where t 2 Œ0; 1/. This means that the extreme points of K are contained in the set ¹e t W t 0º. Since K is the closed convex hull of its extreme points, we conclude that there exists a measure supported in Œ0; 1/ such that Œ0; 1/ D 1 and Z f. dt /; f 2 K: Œ0;1/ This proves (ii) for f 2 Cb . 2, the representing measure is unique. 4. In this case, fa . / WD f . dt / for all a > 0: f . dt / does not depend on a > 0. dt /, we get the representation claimed in (ii) for all f 2 Cb .
4 (Schoenberg). For a function f W S ! C the following assertions are equivalent. (i) f is negative deﬁnite. e. 0/ 0 and s 7! s/ is positive deﬁnite for all t > 0. Proof. bj k / is the n n matrix C with entries aj k bj k . If A; B are non-negative deﬁnite hermitian matrices, then C is non-negative deﬁnite hermitian. aj k //. sj C sk / Á j;kD1;:::;n is a non-negative deﬁnite hermitian matrix. sj / 2 C. This proves that e f is positive deﬁnite. Replacing f by tf , t > 0, the same argument shows that e tf is positive deﬁnite.