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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2509.17902 |
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| _version_ | 1866915953633132544 |
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| author | Choudhury, Projesh Nath Yadav, Shivangi |
| author_facet | Choudhury, Projesh Nath Yadav, Shivangi |
| contents | Entrywise functions preserving positivity and related notions have a rich history, beginning with the seminal works of Schur, Pólya-Szegő, Schoenberg, and Rudin. Following their classical results, it is well-known that entrywise functions preserving positive semidefiniteness for matrices of all dimensions must be real analytic with non-negative Taylor coefficients. These works were taken forward in the last decade by Belton, Guillot, Khare, Putinar, and Rajaratnam. Recently, Belton-Guillot-Khare-Putinar [J. d'Analyse Math. 2023] classified all functions that entrywise preserve totally positive (TP) and totally non-negative (TN) matrices. In this paper, we study entrywise preservers of strictly sign regular and sign regular matrices - a class that includes TP/TN matrices as special cases and was first studied by Schoenberg in 1930 to characterize variation diminution. Our main results provide complete characterizations of entrywise transforms of rectangular matrices which preserve: (i)~sign regularity and strict sign regularity, as well as (ii)~sign regularity and strict sign regularity with a given sign pattern. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17902 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Entrywise preservers of sign regularity Choudhury, Projesh Nath Yadav, Shivangi Functional Analysis Classical Analysis and ODEs Rings and Algebras Entrywise functions preserving positivity and related notions have a rich history, beginning with the seminal works of Schur, Pólya-Szegő, Schoenberg, and Rudin. Following their classical results, it is well-known that entrywise functions preserving positive semidefiniteness for matrices of all dimensions must be real analytic with non-negative Taylor coefficients. These works were taken forward in the last decade by Belton, Guillot, Khare, Putinar, and Rajaratnam. Recently, Belton-Guillot-Khare-Putinar [J. d'Analyse Math. 2023] classified all functions that entrywise preserve totally positive (TP) and totally non-negative (TN) matrices. In this paper, we study entrywise preservers of strictly sign regular and sign regular matrices - a class that includes TP/TN matrices as special cases and was first studied by Schoenberg in 1930 to characterize variation diminution. Our main results provide complete characterizations of entrywise transforms of rectangular matrices which preserve: (i)~sign regularity and strict sign regularity, as well as (ii)~sign regularity and strict sign regularity with a given sign pattern. |
| title | Entrywise preservers of sign regularity |
| topic | Functional Analysis Classical Analysis and ODEs Rings and Algebras |
| url | https://arxiv.org/abs/2509.17902 |