Möbius-transformed trapezoidal rule for polynomial weights
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918475620941824 |
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| author | Hyvönen, Nuutti Suzuki, Yuya |
| author_facet | Hyvönen, Nuutti Suzuki, Yuya |
| contents | This work studies numerical integration by the Möbius-transformed trapezoidal rule, which combines the classical trapezoidal rule with a change of variables induced by a Möbius transformation that maps the unit circle onto the real line. It is shown that this method achieves the optimal convergence rate for a polynomially weighted integral over the real line if the integrand lives in a related polynomially weighted Sobolev space with positive integer smoothness index. This result can also be generalized in a slightly weaker form for fractional smoothness indices via complex interpolation of function spaces. The algorithm only requires pointwise evaluations of the weight and the target integrand at prescribed nodes that do not depend on the integrand and weight in question. The established theoretical convergence rates are verified by numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27758 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Möbius-transformed trapezoidal rule for polynomial weights Hyvönen, Nuutti Suzuki, Yuya Numerical Analysis 65D30, 65D32 This work studies numerical integration by the Möbius-transformed trapezoidal rule, which combines the classical trapezoidal rule with a change of variables induced by a Möbius transformation that maps the unit circle onto the real line. It is shown that this method achieves the optimal convergence rate for a polynomially weighted integral over the real line if the integrand lives in a related polynomially weighted Sobolev space with positive integer smoothness index. This result can also be generalized in a slightly weaker form for fractional smoothness indices via complex interpolation of function spaces. The algorithm only requires pointwise evaluations of the weight and the target integrand at prescribed nodes that do not depend on the integrand and weight in question. The established theoretical convergence rates are verified by numerical experiments. |
| title | Möbius-transformed trapezoidal rule for polynomial weights |
| topic | Numerical Analysis 65D30, 65D32 |
| url | https://arxiv.org/abs/2604.27758 |