Invertible Complex Measures on Euclidean Spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913889062486016 |
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| author | Berger, David Lindner, Alexander |
| author_facet | Berger, David Lindner, Alexander |
| contents | In 1971 Taylor characterised all complex measures on $\mathbb{R}$ that are invertible with respect to convolution as those which can be written in the form $δ_γ\ast σ^{\ast m} \ast \exp(ν)$ for some $γ\in \mathbb{R}$, some complex measure $ν$, some $m\in \mathbb{Z}$ and a given fixed invertible finite signed measure $σ$ (which has characteristic function $\mathbb{R} \ni z \mapsto (1+i z)/(1-i z)$). We extend Taylor's result to complex measures on $\mathbb{R}^n$. Somewhat surprisingly, the structure of invertible complex measures on $\mathbb{R}^n$ is not much more complicated than that of complex measures on $\mathbb{R}$, in the sense that they can be represented as $δ_γ\ast σ_1^{\ast m_1} \ast \ldots \ast σ_p^{\ast m_p} \ast \exp(ν)$ for some $γ\in \mathbb{R}^n$, some complex measure $ν$ and $m_1,\ldots, m_p\in \mathbb{Z}$, where the $σ_i$ correspond to $σ$ in the one-dimensional case and actually live on $1$-dimensional subspaces of $\mathbb{R}^n$. Our proof relies on a general result of Taylor for invertible complex measures on locally compact abelian groups. To apply Taylor's result, we extend some existing results for $\mathbb{C}$-valued functions to functions with values in a semisimple commutative unital Banach algebra with connected Gelfand space. The study of invertible complex measures on $\mathbb{R}^n$ has some impact on the theory of quasi-infinitely divisible probability distributions on $\mathbb{R}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Invertible Complex Measures on Euclidean Spaces Berger, David Lindner, Alexander Probability Functional Analysis In 1971 Taylor characterised all complex measures on $\mathbb{R}$ that are invertible with respect to convolution as those which can be written in the form $δ_γ\ast σ^{\ast m} \ast \exp(ν)$ for some $γ\in \mathbb{R}$, some complex measure $ν$, some $m\in \mathbb{Z}$ and a given fixed invertible finite signed measure $σ$ (which has characteristic function $\mathbb{R} \ni z \mapsto (1+i z)/(1-i z)$). We extend Taylor's result to complex measures on $\mathbb{R}^n$. Somewhat surprisingly, the structure of invertible complex measures on $\mathbb{R}^n$ is not much more complicated than that of complex measures on $\mathbb{R}$, in the sense that they can be represented as $δ_γ\ast σ_1^{\ast m_1} \ast \ldots \ast σ_p^{\ast m_p} \ast \exp(ν)$ for some $γ\in \mathbb{R}^n$, some complex measure $ν$ and $m_1,\ldots, m_p\in \mathbb{Z}$, where the $σ_i$ correspond to $σ$ in the one-dimensional case and actually live on $1$-dimensional subspaces of $\mathbb{R}^n$. Our proof relies on a general result of Taylor for invertible complex measures on locally compact abelian groups. To apply Taylor's result, we extend some existing results for $\mathbb{C}$-valued functions to functions with values in a semisimple commutative unital Banach algebra with connected Gelfand space. The study of invertible complex measures on $\mathbb{R}^n$ has some impact on the theory of quasi-infinitely divisible probability distributions on $\mathbb{R}^n$. |
| title | Invertible Complex Measures on Euclidean Spaces |
| topic | Probability Functional Analysis |
| url | https://arxiv.org/abs/2506.09012 |