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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2604.10394 |
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| _version_ | 1866908957178593280 |
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| author | Graven, Andrew |
| author_facet | Graven, Andrew |
| contents | This paper studies plane domains satisfying a quadrature identity with respect to the singular weight $ρ_0(w)=|w|^{-2}$. These are referred to as log-weighted quadrature domains (LQDs). The logarithmic singularity at $w=0$ leads to phenomena not present in the classical theory: in particular, when the domain contains the origin, the associated quadrature data are no longer unique, but are determined only up to a point charge at $0$. A generalized Schwarz function characterization of LQDs is established together with a natural formulation of the inverse problem in the singular setting. In the simply connected case, it is shown that a domain is an LQD if and only if the outer factor of its Riemann map extends to the exponential of a rational function. This characterization yields explicit formulae relating the quadrature function and the Riemann map via the Faber transform, thereby extending earlier formulae from the non-singular theory. Several basic classes of LQDs are also covered, and explicit examples are computed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10394 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analysis of Log-Weighted Quadrature Domains Graven, Andrew Complex Variables Numerical Analysis Mathematical Physics Analysis of PDEs 31A25 (Primary) 30E25, 30C20, 31A15 (Secondary) This paper studies plane domains satisfying a quadrature identity with respect to the singular weight $ρ_0(w)=|w|^{-2}$. These are referred to as log-weighted quadrature domains (LQDs). The logarithmic singularity at $w=0$ leads to phenomena not present in the classical theory: in particular, when the domain contains the origin, the associated quadrature data are no longer unique, but are determined only up to a point charge at $0$. A generalized Schwarz function characterization of LQDs is established together with a natural formulation of the inverse problem in the singular setting. In the simply connected case, it is shown that a domain is an LQD if and only if the outer factor of its Riemann map extends to the exponential of a rational function. This characterization yields explicit formulae relating the quadrature function and the Riemann map via the Faber transform, thereby extending earlier formulae from the non-singular theory. Several basic classes of LQDs are also covered, and explicit examples are computed. |
| title | Analysis of Log-Weighted Quadrature Domains |
| topic | Complex Variables Numerical Analysis Mathematical Physics Analysis of PDEs 31A25 (Primary) 30E25, 30C20, 31A15 (Secondary) |
| url | https://arxiv.org/abs/2604.10394 |