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

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. We show that in the more dimensional anisotropic case the classical log-convex condition is not sufficient: convexity as a function of more variables is needed. We finally obtain some applications to the inclusion of spaces of rapidly decreasing ultradifferentiable functions in the matrix weighted setting.