Construction of the log-convex minorant of a sequence $\{M_α\}_{α\in\mathbb{N}_0^d}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909492405338112 |
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| author | Boiti, Chiara Jornet, David Oliaro, Alessandro Schindl, Gerhard |
| author_facet | Boiti, Chiara Jornet, David Oliaro, Alessandro Schindl, Gerhard |
| contents | We give a simple construction of the log-convex minorant of a sequence $\{M_α\}_{α\in\mathbb{N}_0^d}$ 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 $ω_M$, that is $M_p=\sup_{t>0}t^p\exp(-ω_M(t))$. We show that in the more dimensional anisotropic case the classical log-convex condition $M_α^2\leq M_{α-e_j}M_{α+e_j}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_11245 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction of the log-convex minorant of a sequence $\{M_α\}_{α\in\mathbb{N}_0^d}$ Boiti, Chiara Jornet, David Oliaro, Alessandro Schindl, Gerhard Functional Analysis 40B05, 46A13, 46A45 We give a simple construction of the log-convex minorant of a sequence $\{M_α\}_{α\in\mathbb{N}_0^d}$ 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 $ω_M$, that is $M_p=\sup_{t>0}t^p\exp(-ω_M(t))$. We show that in the more dimensional anisotropic case the classical log-convex condition $M_α^2\leq M_{α-e_j}M_{α+e_j}$ 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. |
| title | Construction of the log-convex minorant of a sequence $\{M_α\}_{α\in\mathbb{N}_0^d}$ |
| topic | Functional Analysis 40B05, 46A13, 46A45 |
| url | https://arxiv.org/abs/2401.11245 |