Integral representations of continuous functions (Q1179959)

The main result is given as Theorem 1. A continuous function \(\sigma(\lambda)\), \(\lambda\geq 0\), has a representation \[ \sigma(\lambda)=\sum^{n-1}_{i=0}a_ i\lambda^ 2+\int^ \infty_ 0\left[e^{-\lambda u}-1-(1+u^ n)^{-1}\sum^{n- 1}_{i=1}((-\lambda u)^ i/i!)\right](1-e^{-u})^{-n}G(du), \] where \(n\) is a positive integer, the \(a_ i\) \((i=0,1,\dots,n-1)\) are constants, \(G\) is a finite measure on \([0,\infty)\) and the value of the integrand at \(u=0\) is defined by continuity as \(((-\lambda)^ n/n!)\); iff for \(c\geq 0\) the function \(\theta_ c(\lambda):=(-1)^ n\Delta^ n_ c\sigma(\lambda)\), \(\lambda\geq 0\) is completely monotone. Two lemmas lead to the proof of the above theorem as well as two corollaries emanate from the theorem for special values of \(n\).