Construction of the log‐convex minorant of a sequence {Mα}α∈N0d$\lbrace M_\alpha \rbrace _{\alpha \in \mathbb {N}_0^d}$

Abstract

We give a simple construction of the log‐convex minorant of a sequence and consequently extend to the ‐dimensional case the well‐known formula that relates a log‐convex sequence to its associated function , that is, . 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 (not only coordinate‐wise). We finally obtain some applications to the inclusion of spaces of rapidly decreasing ultradifferentiable functions in the matrix weighted setting.