Multivariate transforms of total positivity
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915039944900608 |
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| author | Damase, Sujit Sakharam Khare, Apoorva |
| author_facet | Damase, Sujit Sakharam Khare, Apoorva |
| contents | Belton-Guillot-Khare-Putinar [J. d'Analyse Math. 2023] classified the post-composition operators that preserve TP/TN kernels of each specified order. We explain how to extend this from preservers to transforms, and from one to several variables. Namely, given arbitrary nonempty totally ordered sets $X,Y$, we characterize the transforms that send each tuple of kernels on $X \times Y$ that are TP/TN of orders $k_1, \dots, k_p$, to a TP/TN kernel of order $l$, for arbitrary positive integers (or infinite) $k_j$ and $l$. An interesting feature is that to preserve TP (or TN) of order $2$, the preservers are products of individual power (or Heaviside) functions in each variable; but for all higher orders, the preservers are powers in a single variable. We also classify the multivariate transforms of symmetric TP/TN kernels; in this case it is the preservers of TP/TN of order 3 that are multivariate products of power functions, and of order 4 that are individual powers. The proofs use generalized Vandermonde kernels, Hankel kernels, (strictly totally positive) Polya frequency functions, and a kernel studied recently but tracing back to works of Schoenberg [Ann. of Math. 1955] and Karlin [Trans. Amer. Math. Soc. 1964]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03391 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multivariate transforms of total positivity Damase, Sujit Sakharam Khare, Apoorva Functional Analysis Classical Analysis and ODEs 15B48 (primary), 15A15, 30C40, 39B62, 47B34 (secondary) Belton-Guillot-Khare-Putinar [J. d'Analyse Math. 2023] classified the post-composition operators that preserve TP/TN kernels of each specified order. We explain how to extend this from preservers to transforms, and from one to several variables. Namely, given arbitrary nonempty totally ordered sets $X,Y$, we characterize the transforms that send each tuple of kernels on $X \times Y$ that are TP/TN of orders $k_1, \dots, k_p$, to a TP/TN kernel of order $l$, for arbitrary positive integers (or infinite) $k_j$ and $l$. An interesting feature is that to preserve TP (or TN) of order $2$, the preservers are products of individual power (or Heaviside) functions in each variable; but for all higher orders, the preservers are powers in a single variable. We also classify the multivariate transforms of symmetric TP/TN kernels; in this case it is the preservers of TP/TN of order 3 that are multivariate products of power functions, and of order 4 that are individual powers. The proofs use generalized Vandermonde kernels, Hankel kernels, (strictly totally positive) Polya frequency functions, and a kernel studied recently but tracing back to works of Schoenberg [Ann. of Math. 1955] and Karlin [Trans. Amer. Math. Soc. 1964]. |
| title | Multivariate transforms of total positivity |
| topic | Functional Analysis Classical Analysis and ODEs 15B48 (primary), 15A15, 30C40, 39B62, 47B34 (secondary) |
| url | https://arxiv.org/abs/2411.03391 |