On the regularization of sequences and associated weight functions

TL;DR

This paper revisits and extends the geometric regularization of sequences using weight functions, crucial for ultradifferentiable function classes, including non-standard cases and blow-up scenarios.

Contribution

It generalizes Mandelbrojt's regularization method, accommodating non-standard situations and allowing blow-up in the regularizing function, enhancing the theory of ultradifferentiable classes.

Findings

Extended the regularization procedure to non-standard cases.

Allowed blow-up in the regularizing function.

Provided new insights into convex minorants of sequences.

Abstract

We revisit and generalize the geometric procedure of regularizing a sequence of real numbers with respect to a so-called regularizing function. This approach was studied by S. Mandelbrojt and becomes useful and necessary when working with corresponding classes of ultradifferentiable functions defined via weight sequences and analogous weighted spaces. In this note we also study non-standard situations for the construction yielding the (log-)convex minorant of a sequence and allow a "blow-up" for the regularizing function.