Summation of formal power series through iterated Laplace integrals. (English) Zbl 0769.34004

The author defines a summability method of formal power series through iterated Laplace integrals, which is denoted as \((\kappa_ 1,\dots,\kappa_ p)-iL\) summability, and gives three conditions for such a formal power series to be \((\kappa_ 1,\dots,\kappa_ p)-iL\) summable in a direction \(d\), \(d\) being an arbitrary real number. He gives an example of a series \(\widehat f=\sum_{n=0}^ \infty f_ n z^ n\), which is \((1,1)-iL\)-summable, but not \(k\)-summable in any direction \(d\) for arbitrary \(k>0\), that means, that \(\widehat f\) corresponds uniquely to a function \(f\) which is analytic in a sector \(\{z\): \(| z|>0\), \(d-\alpha/2<\arg z<d+\alpha/2\}\) on the Riemann surface of \(\log z\) and satisfies an estimate \(| f(z)-\sum_{n=0}^ N f_ n z^ n|\leq CK^ N| z|^{N\Gamma(1+N/k)}\).
Reviewer: F.Rühs (Freiberg)


34M99 Ordinary differential equations in the complex domain
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
30B10 Power series (including lacunary series) in one complex variable
40C15 Function-theoretic methods (including power series methods and semicontinuous methods) for summability
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