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Entrywise functions preserving Loewner positivity have been studied by manyauthors, most notably Schoenberg and Rudin. Following their work, it is knownthat functions preserving positivity when applied entrywise to positivesemidefinite matrices of all dimensions are necessarily analytic withnonnegative Taylor coefficients. When the dimension is fixed, it has been shownby Vasudeva and Horn that such functions are automatically continuous andsufficiently differentiable. A natural refinement of the aforementioned problemconsists of characterizing functions preserving positivity under rankconstraints. In this paper, we begin this study by characterizing entrywisefunctions which preserve the cone of positive semidefinite real matrices ofrank $1$ with entries in a general interval. Classifying such functions isintimately connected to the classical problem of solving Cauchy's functionalequations, which have non-measurable solutions. We demonstrate that under mildlocal measurability assumptions, such functions are automatically smooth andcan be completely characterized. We then extend our results by classifyingfunctions preserving positivity on rank $1$ Hermitian complex matrices.

Loewner positive entrywise functions, and classification of measurable solutions of Cauchy's functional equations