Construction of the log-convex minorant of a sequence {M _α} _α∈N0d

We give a simple construction of the log-convex minorant of a sequence (Formula presented.) and consequently extend to the (Formula presented.) -dimensional case the well-known formula that relates a log-convex sequence (Formula presented.) to its associated function (Formula presented.), that is, (Formula presented.). We show that in the more dimensional anisotropic case the classical log-convex condition (Formula presented.) 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.