Non-negative Solutions
N.A. Watson · 2017
In this chapter, we establish the fundamental result that every nonnegative temperature on R̲ n × ]0, a[ can be represented as the Gauss-Weierstrass integral of a non-negative measure on R̲ n . 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 R̲ n × ]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 R̲ n × ]0, a[ can decay as t → 0 + or ‖x‖ → ∞.