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A panorama of positivity. I: dimension free

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Publication date31/05/2019
Host publicationAnalysis of Operators on Function Spaces: The Serguei Shimorin Memorial Volume
EditorsAlexandru Aleman, Haakan Hedenmalm, Dmitry Khavinson, Mihai Putinar
Place of PublicationCham
Number of pages49
ISBN (electronic)9783030146405
ISBN (print)9783030146399
<mark>Original language</mark>English

Publication series

NameTrends in Mathematics
ISSN (Print)2297-0215
ISSN (electronic)2297-024X


This survey contains a selection of topics unified by the concept of positive semidefiniteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical references. Modern applications to high-dimensional covariance estimation and regularization are included.