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Bernstein’s analyticity theorem for quantum differences

Tord Sjödin (2007)

Czechoslovak Mathematical Journal

We consider real valued functions f defined on a subinterval I of the positive real axis and prove that if all of f ’s quantum differences are nonnegative then f has a power series representation on I . Further, if the quantum differences have fixed sign on I then f is analytic on I .

Bounds for quotients in rings of formal power series with growth constraints

Vincent Thilliez (2002)

Studia Mathematica

In rings Γ M of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence M = ( M l ) l 0 (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in Γ M such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to Γ M , provided Γ M is stable under derivation. By a further development of the method, we obtain the main result of the paper, stating that...

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