Preliminary Note on the Inversion of the Laplace Integral

VOL. 18, 1932

MA THEMA TICS: D. V. WIDDER

181

PRELIMINARY NOTE ON THE INVERSION OF THE LAPLACE INTEGRAL By D. V. WIDDER DEPARTMENT

OF

MATHEMATICS, HARVARD UNIVERSITY

Communicated January 15, 1932

In a previous note in these PROCEEDINGS' we stated the following result: If (p(t) is continuous in the interval 0 < t < o, if lim (p(t) = a, and if 1=

X

co

f(x) = J'ext1p(t)dt, then kim [x^ +I ( 1)kf (,x) -

ip(p

)]=°

(1)

uniformly in the interval 0 _ x < a. This result was obtained in order to discuss the zeros of gp(t) in terms of those of the derivatives of f(x). The result has, however, an important interest in itself, since it enables us to invert the Laplace integral (1) under the conditions described in the theorem. We have, in fact,

(o(t)

=

urn

(_ 1)k f(k) (k/t)k

+1

uniformly for 0 < t < c. This result was obtained earlier by E. Post2 under the restriction of continuity on sp(t) but without the condition that p(t) approaches a limit as t becomes infinite. On the other hand the uniformity of the approach was not obtained by Post. In the present note we sketch a theory whereby the integral equation (1) may be solved with no restriction on so(t) as to continuity. We simply impose the natural condition that s(t) shall be integrable in the sense of Lebesgue and (so that the integral (1) may converge for x sufficiently large) the condition Ip(t)

<Mea

0O c is that

f(x) =

,co e"-x da(t),

where a (t) is non-decreasing and the integral converges for x > c. THEOREM 8. A necessary and suicient condition thai f(x) can be represented in the form C*

f (x)

=

e-Xtda(t),

the integral converging absolutely for x > 0, is that Jc

uk-| f(k+l)(U)

I du < M

x > 0; k = O, 1, 2.

For earlier proofs of Theorem 7 and Theorem 8 reference may be made to the author's paper already cited.3 The proofs now obtained have distinct advantages over those given earlier in that they involve no reference to the moment problem. 1 D. V. Widder, "On the Changes of Sign of the Derivatives of a Function Defined by a Laplace Integral," Proc. Nat. Acad. Sci., 18, 112-114(1932). 2 E. L. Post, "Generalized Differentiation," Trans. Am. Math. Soc., 32, 723 (1930). 3 D. V. Widder, "Necessary and Sufficient Conditions for the Representation of a Function as a Laplace Integral," Trans. Am. Math. Soc., 33, 851 (1931).