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Consider an expansion of the real field in which every unary definable continuous function can be ultimately majorized by a definable analytic function. We prove the strict Positivstellensatz for analytic functions which are definable in such structures. The methods also work for a large class of quasianalytic subrings of the ring of those smooth functions that are definable in a polynomially bounded structure.

A strict Positivstellensatz for rings of definable analytic functions