A continual analogue of a theorem by M. Fekete and G. Pólya (Q5928180)

Let \(p\) be continuous in \([a,b]\) with \(p(a)> 0\), \(p(b)> 0\), and \(L(x,p)= \int^b_a e^{xt}p(t) dt> 0\) for all real \(x\). The authors prove that for suitable small \(\varepsilon\) and for suitable large \(\lambda\) the product \(\exp(\lambda e^{\varepsilon x})L(x,p)\) is the Laplace transform of a nonnegative function. Also, a more complicated version is treated in the case that \(a=0\) and \(b= \infty\).