A Striktpositivstellensatz for measurable functions (corrected version)

TL;DR

This paper extends the sums of squares decomposition to positive Borel measurable functions on bounded sets, using duality and spectral theory, broadening the scope of previous polynomial-focused results.

Contribution

It introduces a new Striktpositivstellensatz for measurable functions, generalizing polynomial results to a broader class of functions via spectral methods.

Findings

Decomposition of positive measurable functions achieved

Utilizes duality and spectral theorem techniques

Broadens applications of sums of squares methods

Abstract

A weighted sums of squares decomposition of positive Borel measurable functions on a bounded Borel subset of the Euclidean space is obtained via duality from the spectral theorem for tuples of commuting self-adjoint operators. The analogous result for polynomials or certain rational functions was amply exploited during the last decade in a variety of applications.