ABSTRACT
In this chapter, we establish the fundamental result that every nonnegative temperature on R̲n × ]0, a[ can be represented as the Gauss-Weierstrass integral of a non-negative measure on R̲n. This involves a notion of convergence of a sequence of measures, as well as the maximum principle on a circular cylinder and an extension of this to R̲n × ]0, a[. This extension, Theorem 2.4, is also essential for the results of Chapter III. To conclude the present chapter, we use the representation theorem to characterize Gauss-Weierstrass integrals and find the greatest rate at which a positive temperature on R̲n × ]0, a[ can decay as t → 0+ or ‖x‖ → ∞.
