Interpolation of Completely Monotone Functions 107

Proof of Lemma a* : Let ß and jc satisfy the assumptions and let /ei^,«. Then, by (6)

\fix ) -/„(x)| ^ d(f,ß)\c„(x)\ f e-'" [e{x,-)*Kmdt =

0

= —~~Cn{x,ß), x + ß

Proof of Lemma b* : Let/9, x and/satisfy the assumptions. Rewrite (6) in the form

Ifnix ) -/(x)| = f e^' Je-(^+«<'-'>rfG„(v)^(0, (7)

0 0

where G„ (0 = | c„ (x) | J g„ (v) dv and g„ is the convolution of functions

0

e { Xj + ß,-), 7=1,2,...,w. It follows from (3) that the Laplace

00

transforms C„(x,^ + 5) = J e ~ ^^ofG„ (/) converge uniformly on [0, d]

0 as « -> 00 and therefore G„ converge to a non-decreasing function G. Using the fact that the functions g„ are bounded and strictly positive on (0, 00) we can show easily that G is continuous and strictly positive on (0, 00). Applying the Helly theorem to the inner integral in (7) and the dominated convergence theorem to the outer integral, we see that the right-hand side of (7) converges to a similar integral with G„ replaced by G, and this integral is strictly positive.

Proof of Theorem 1* : Let x > — a, x # A,. There exists ^8 < a such that X > — ß, Xj> — ß for all j and, under the assumptions of Theorem 1*, (3) holds for any finite (5 > 0, so that Lemma b* appUes.

Proof of Theorem 2* ; Put Я^ = Ao for all;. If x > a -f 2 Яо then there exists ß>0L such that х>/?Ч-2Яо and for this ß,Cn(x,ß)-^ 00. Hence, Lemma a* applies.

References

[ 1 ] Feller, W. : On Müntz' theorem and completely monotone functions. Amer. Math. Monthly 75, 342—350 (1968).

M . JIRINA

School of Mathematical Sciences

The Flinders University of South Australia

Bedford Park, S. A. 5042, AustraUa

8 Monatshefte fur Mathemaük, Bd 94/2