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Two-Dimensional Diffusion Orthogonal Polynomials Ordered by a Weighted Degree

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Abstract

We study the problem of describing the triples \((\Omega,g,\mu)\), \(\mu=\rho\,dx\), where \(g= (g^{ij}(x))\) is the (co)metric associated with a symmetric second-order differential operator \(\mathbf{L}(f) = \frac{1}{\rho}\sum_{ij} \partial_i (g^{ij} \rho\,\partial_j f)\) defined on a domain \(\Omega\) of \(\mathbb{R}^d\) and such that there exists an orthonormal basis of \(\mathcal{L}^2(\mu)\) consisting of polynomials which are eigenvectors of \(\mathbf{L}\) and this basis is compatible with the filtration of the space of polynomials by some weighted degree.

In a joint paper of D. Bakry, M. Zani, and the author this problem was solved in dimension 2 for the usual degree. In the present paper we solve it still in dimension 2 but for a weighted degree with arbitrary positive weights.

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Table 1.

Notes

  1. There is also a misprint in this theorem: \(X^2\) and \(Y^2\) should be interchanged in \(G\).

References

  1. D. Bakry and X. Bressaud, “Diffusions with polynomial eigenvectors via finite subgroups of \(O(3)\)”, Ann. Fac. Sci. Toulouse Math. (6), 25:2–3 (2016), 683–721.

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Funding

Supported by grant RNF No. 19-11-00316.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Moscow, Russia

    S. Yu. Orevkov

Authors
  1. S. Yu. Orevkov
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Correspondence to S. Yu. Orevkov.

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Additional information

Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2023, Vol. 57, pp. 39–73 https://doi.org/10.4213/faa4012.

Translated by S. Yu. Orevkov

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Orevkov, S.Y. Two-Dimensional Diffusion Orthogonal Polynomials Ordered by a Weighted Degree. Funct Anal Its Appl 57, 208–235 (2023). https://doi.org/10.1134/S0016266323030036

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