Construction of the log-convex minorant of a sequence $\{M_{\alpha }{\}}_{\alpha \in {\mathbb{N}}_0^d}$

Abstract

We give a simple construction of the log-convex minorant of a sequence $\{M_{\alpha }{\}}_{\alpha \in {\mathbb{N}}_0^d}$ and consequently extend to the $d$-dimensional case the well-known formula that relates a log-convex sequence $\{M_p{\}}_{p\in {\mathbb{N}}_0}$ to its associated function ${\omega }_M$, that is $M_p={\sup }_{t>0}{t}^p\exp (-{\omega }_M(t))$. We show that in the more dimensional anisotropic case the classical log-convex condition $M_{\alpha }^2\le M_{\alpha -e_j}{M}_{\alpha +e_j}$ is not sufficient: convexity as a function of more variables is needed (not only coordinate-wise). We finally obtain some applications to the inclusion of spaces of rapidly decreasing ultradifferentiable functions in the matrix weighted setting.

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