On inversion of absolutely convergent weighted Dirichlet series in two variables
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911971110027264 |
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| author | Dabhi, Prakash A. |
| author_facet | Dabhi, Prakash A. |
| contents | Let $0<p\leq 1$, and let $ω:\mathbb N^2 \to [1,\infty)$ be an almost monotone weight. Let $\mathbb H$ be the closed right half plane in the complex plane. Let $\widetilde a$ be a complex valued function on $\mathbb H^2$ such that $\widetilde a(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}a(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\in \mathbb H^2$ with $\sum_{(m,n)\in \mathbb N^2} |a(m,n)|^pω(m,n)<\infty$. If $\widetilde a$ is bounded away from zero on $\mathbb H^2$, then there is an almost monotone weight $ν$ on $\mathbb N^2$ such that $1\leq ν\leq ω$, $ν$ is constant if and only if $ω$ is constant, $ν$ is admissible if and only if $ω$ is admissible, the reciprocal $\frac{1}{\widetilde a}$ has the Dirichlet representation $\frac{1}{\widetilde a}(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}b(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\in \mathbb H^2$ and $\sum_{(m,n)\in \mathbb N^2}|b(m,n)|^pν(m,n)<\infty$. If $φ$ is holomorphic on a neighbourhood of the closure of range of $\widetilde a$, then there is an almost monotone weight $ξ$ on $\mathbb N^2$ such that $1\leq ξ\leq ω$, $ξ$ is constant if and only if $ω$ is constant, $ξ$ is admissible if and only if $ω$ is admissible, $φ\circ \widetilde a$ has the Dirichlet series representation $(φ\circ \widetilde a)(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2} c(m,n)m^{-s_1}n^{-s_2}\;((s_1,s_2)\in \mathbb H^2)$ and $\sum_{(m,n)\in \mathbb N^2}|c(m,n)|^pξ(m,n)<\infty$. Let $ω$ be an admissible weight on $\mathbb N^2$, and let $\widetilde a$ have $p$-th power $ω$- absolutely convergent Dirichlet series. Then it is shown that the reciprocal of $\widetilde a$ has $p$-th power $ω$- absolutely convergent Dirichlet series if and only if $\widetilde a$ is bounded away from zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19982 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On inversion of absolutely convergent weighted Dirichlet series in two variables Dabhi, Prakash A. Functional Analysis Primary 30B50, Secondary 30B50, 46H99, 46J05 Let $0<p\leq 1$, and let $ω:\mathbb N^2 \to [1,\infty)$ be an almost monotone weight. Let $\mathbb H$ be the closed right half plane in the complex plane. Let $\widetilde a$ be a complex valued function on $\mathbb H^2$ such that $\widetilde a(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}a(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\in \mathbb H^2$ with $\sum_{(m,n)\in \mathbb N^2} |a(m,n)|^pω(m,n)<\infty$. If $\widetilde a$ is bounded away from zero on $\mathbb H^2$, then there is an almost monotone weight $ν$ on $\mathbb N^2$ such that $1\leq ν\leq ω$, $ν$ is constant if and only if $ω$ is constant, $ν$ is admissible if and only if $ω$ is admissible, the reciprocal $\frac{1}{\widetilde a}$ has the Dirichlet representation $\frac{1}{\widetilde a}(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}b(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\in \mathbb H^2$ and $\sum_{(m,n)\in \mathbb N^2}|b(m,n)|^pν(m,n)<\infty$. If $φ$ is holomorphic on a neighbourhood of the closure of range of $\widetilde a$, then there is an almost monotone weight $ξ$ on $\mathbb N^2$ such that $1\leq ξ\leq ω$, $ξ$ is constant if and only if $ω$ is constant, $ξ$ is admissible if and only if $ω$ is admissible, $φ\circ \widetilde a$ has the Dirichlet series representation $(φ\circ \widetilde a)(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2} c(m,n)m^{-s_1}n^{-s_2}\;((s_1,s_2)\in \mathbb H^2)$ and $\sum_{(m,n)\in \mathbb N^2}|c(m,n)|^pξ(m,n)<\infty$. Let $ω$ be an admissible weight on $\mathbb N^2$, and let $\widetilde a$ have $p$-th power $ω$- absolutely convergent Dirichlet series. Then it is shown that the reciprocal of $\widetilde a$ has $p$-th power $ω$- absolutely convergent Dirichlet series if and only if $\widetilde a$ is bounded away from zero. |
| title | On inversion of absolutely convergent weighted Dirichlet series in two variables |
| topic | Functional Analysis Primary 30B50, Secondary 30B50, 46H99, 46J05 |
| url | https://arxiv.org/abs/2407.19982 |