The multivariate Herglotz-Nevanlinna class: Rational approximation
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909797583945728 |
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| author | Bhowmik, Mainak Putinar, Mihai |
| author_facet | Bhowmik, Mainak Putinar, Mihai |
| contents | We return to Takagi's variational principle, generalized after forty years to two complex variables by Pfister. Both isolating some extremal rational functions associated to a bounded holomorphic function in the unit disk, respectively the bidisk. The rational inner functions arising from the Takagi-Pfister skew eigenvectors lead to a Pade type approximation scheme. For these rational functions, we prove a Montessus de Ballore type convergence theorem, on the polydisk in any complex dimension. On the natural and more restrictive class of Agler holomorphic functions with non-negative real part, we show that Cayley rational inner functions match any finite section of the Taylor expansion at a prescribed point. We derive from the Hilbert space proof that the finite section coefficient set of Taylor series of the Agler functions in the Herglotz-Nevanlinna setting is semi-algbraic. The pole distribution of the Takagi-Pfister interpolation sequence is identified as a main open question on the subject. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15668 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The multivariate Herglotz-Nevanlinna class: Rational approximation Bhowmik, Mainak Putinar, Mihai Complex Variables Functional Analysis 32E30, 32A17, 47A48, 41A05, 41A20, 14P10 We return to Takagi's variational principle, generalized after forty years to two complex variables by Pfister. Both isolating some extremal rational functions associated to a bounded holomorphic function in the unit disk, respectively the bidisk. The rational inner functions arising from the Takagi-Pfister skew eigenvectors lead to a Pade type approximation scheme. For these rational functions, we prove a Montessus de Ballore type convergence theorem, on the polydisk in any complex dimension. On the natural and more restrictive class of Agler holomorphic functions with non-negative real part, we show that Cayley rational inner functions match any finite section of the Taylor expansion at a prescribed point. We derive from the Hilbert space proof that the finite section coefficient set of Taylor series of the Agler functions in the Herglotz-Nevanlinna setting is semi-algbraic. The pole distribution of the Takagi-Pfister interpolation sequence is identified as a main open question on the subject. |
| title | The multivariate Herglotz-Nevanlinna class: Rational approximation |
| topic | Complex Variables Functional Analysis 32E30, 32A17, 47A48, 41A05, 41A20, 14P10 |
| url | https://arxiv.org/abs/2509.15668 |