Zeros and regular growth of Laplace transforms along curves. (English) Zbl 0733.44001

Zeinstra, Rein

J. Reine Angew. Math. 424, 1-15 (1992).

We extend some results of Müntz-Szász type due to Korevaar and Leont’ev: Let \(\gamma (t)=t+i\) h(t), \(0\leq t<\infty\), be a Lipschitz curve and let \((\lambda_ n)\) be an increasing sequence of positive numbers such that \(\sum \lambda_ n^{-1}=\infty\). Then, under a technical additional condition on \(\gamma\) which is satisfied if h(t) is piecewise \(C^ 1\) or piecewise monotone, the exponentials \(\exp (- \lambda_ nz)\) span \(C_ 0(\gamma)\). More generally, a Blaschke type convergence condition is given for the zeros of Laplace transforms of measures supported on \(\gamma\), and it is shown that these Laplace transforms are of very regular growth in the sense of Ahlfors-Heins- Azarin.

Reviewer: R.Zeinstra

MSC:

44A10

Laplace transform

Keywords:

zeros; regular growth; Laplace transforms along curves; Blaschke convergence; Ahlfors-Heins-Azarin growth; Müntz-Szász theorem