Two BPHZ convergence theorems are proved directly in Euclidean position space, without exponentiating the propagators, making use of the Cluster Convergence Theorem presented previously. The first theorem proves the absolute convergence of arbitrary BPHZ-renormalized Feynman diagrams, when counterterms are allowed for one-linereducible subdiagrams, as well as for one-line-irreducible subdiagram...

In this paper, a strong convergence theorem for nonexpansive mappings in a uniformly convex and smooth Banach space is proved without using Bruck’s result, see RE Bruck (Bruck 1981). This theorem is different from the recent strong convergence theorems due to S Matsushita and W Takahashi (Matsushita and Takahashi 2008) and HK Xu (Xu 2006).

We will also use Fubini’s theorem [2, p. 170–171] and Lebesgue’s dominated convergence theorem [2, p. 91]. Fubini’s theorem allows us to change the order of integration in a double integral of a function, when we know that the function is integrable in one of the orders (i.e., the integral of the absolute value of the function is finite in one of the orders). Lebesgue’s dominated convergence th...