On the regularization of sequences and associated weight functions

We revisit and generalize the geometric procedure of regularizing a sequence of real numbers with respect to a so-called regularizing function. This approach is due to 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.