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Totally positive kernels, Pólya frequency functions, and their transforms

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Abstract

The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A. M. Whitney’s density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Pólya frequency functions, and Pólya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.

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Acknowledgments

We thank Percy Deift for raising the question of classifying total positivity preservers and pointing out the relevance of such a result for current studies in mathematical physics. We also thank Alan Sokal for his comments on a preliminary version of the paper. A. B. is grateful for the hospitality of the Indian Institute of Science, Bangalore, where part of this work was carried out. D. G. was partially supported by a University of Delaware Research Foundation grant, by a Simons Foundation collaboration grant for mathematicians, and by a University of Delaware strategic initiative grant. A. K. was partially supported by Ramanujan Fellowship grant SB/S2/RJN-121/2017,MATRICS grant MTR/2017/000295, and SwarnaJayanti Fellowship grants SB/SJF/2019-20/14 and DST/SJF/MS/2019/3 from SERB and DST (Govt. of India), by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India), and by a Young Investigator Award from the Infosys Foundation. M. P. was partially supported by a Simons Foundation collaboration grant for mathematicians.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Lancaster University, LA1 4YW, Lancaster, UK

    Alexander Belton

  2. Department of Mathematical Sceinces, University of Delaware, Newark, DE, 19716, USA

    Dominique Guillot

  3. Department of Mathematics and Analysis and Probability Research Group, Indian Institute of Science, 560012, Bangalore, India

    Apoorva Khare

  4. Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA, 93106-3080, USA

    Mihai Putinar

  5. School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

    Mihai Putinar

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  1. Alexander Belton
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Correspondence to Apoorva Khare.

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Belton, A., Guillot, D., Khare, A. et al. Totally positive kernels, Pólya frequency functions, and their transforms. JAMA 150, 83–158 (2023). https://doi.org/10.1007/s11854-022-0259-7

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