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

TL;DR

This paper introduces a simple method to construct the log-convex minorant of multi-dimensional sequences and extends classical one-dimensional formulas to higher dimensions, with applications to ultradifferentiable function spaces.

Contribution

It provides a new construction for the log-convex minorant in multiple dimensions and clarifies the convexity conditions needed beyond the classical coordinate-wise approach.

Findings

Classical log-convexity condition is insufficient in higher dimensions.

Extended the formula relating sequences to their associated functions to multiple dimensions.

Applied results to ultradifferentiable function space inclusions.

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 $M_\alpha^2\leq M_{\alpha-e_j}M_{\alpha+e_j}$ 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.