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A nonnegativity certificate (NNC) is a way to write a polynomial so that its nonnegativity on a semialgebraic set becomes evident. Positivstellensätze (Psätze) guarantee the existence of NNCs. Both NNCs and Psätze underlie powerful algorithmic techniques for optimization. This paper proposes a universal approach to derive new Psätze for general semialgebraic sets from ones developed for simpler sets, such as a box, a simplex, or the nonnegative orthant. We provide several results illustrating the approach. First, by considering Handelman’s Positivstellensatz (Psatz) over a box, we construct non-SOS Schmüdgen-type Psätze over any compact semialgebraic set, that is, a family of Psätze that follow the structure of the fundamental Schmüdgen’s Psatz but where instead of SOS polynomials, any class of polynomials containing the nonnegative constants can be used, such as SONC, DSOS/SDSOS, hyperbolic, or …