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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.12937 |
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| _version_ | 1866913564483125248 |
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| author | Chuaqui, Martin Hernández, Rodrigo Llinares, Adrián Mas, Alejandro |
| author_facet | Chuaqui, Martin Hernández, Rodrigo Llinares, Adrián Mas, Alejandro |
| contents | We study the convolution of functions of the form \[ f_α(z) := \dfrac{\left( \frac{1 + z}{1 - z} \right)^α- 1}{2 α}, \] which map the open unit disk of the complex plane onto polygons of 2 edges when $α\in(0,1)$. We extend results by Cima by studying limits of convolutions of finitely many $f_α$ and by considering the convolution of arbitrary unbounded convex mappings. The analysis for the latter is based on the notion of angle at infinity, which provides an estimate for the growth at infinity and determines whether the convolution is bounded or not. A generalization to an arbitrary number of factors shows that the convolution of $n$ randomly chosen unbounded convex mappings has a probability of $1/n!$ of remaining unbounded. We also extend Cima's analysis on the coefficients of the functions $f_α$ by providing precise asymptotic behavior for all $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_12937 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the convolution of convex 2-gons Chuaqui, Martin Hernández, Rodrigo Llinares, Adrián Mas, Alejandro Complex Variables 30C45 We study the convolution of functions of the form \[ f_α(z) := \dfrac{\left( \frac{1 + z}{1 - z} \right)^α- 1}{2 α}, \] which map the open unit disk of the complex plane onto polygons of 2 edges when $α\in(0,1)$. We extend results by Cima by studying limits of convolutions of finitely many $f_α$ and by considering the convolution of arbitrary unbounded convex mappings. The analysis for the latter is based on the notion of angle at infinity, which provides an estimate for the growth at infinity and determines whether the convolution is bounded or not. A generalization to an arbitrary number of factors shows that the convolution of $n$ randomly chosen unbounded convex mappings has a probability of $1/n!$ of remaining unbounded. We also extend Cima's analysis on the coefficients of the functions $f_α$ by providing precise asymptotic behavior for all $α$. |
| title | On the convolution of convex 2-gons |
| topic | Complex Variables 30C45 |
| url | https://arxiv.org/abs/2311.12937 |