I will give a gentle introduction to total positivity and the theory of Polya frequency (PF) functions. This includes their spectral properties, basic examples including via convolution, and a few proofs to show how the main ingredients fit together. Many classical results (and one Hypothesis) from before 1955 feature in this journey. I will end by describing how PF functions connect to the Laguerre–Polya class and hence Polya–Schur multipliers, and mention 21st century incarnations of the latter.

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Last updated: 11 Dec 2023