Very slowly varying functions. II

Abstract

Abstract. This paper is a sequel to papers by Ash, Erdős and Rubel, on very slowly varying functions, and by Bingham and Ostaszewski, on foundations of regular variation. We show that generalizations of the Ash-Erdős-Rubel approach-imposing growth restrictions on the function h, rather than regularity conditions such as measurability or the Baire property-lead naturally to the main result of regular variation, the Uniform Convergence Theorem.1. Introduction. We work with the Karamata theory of regular and slow variation; see [BGT] for a monograph account. Here the main result is the Uniform Convergence Theorem (UCT) which asserts that the defining pointwise convergence for slow variation in fact holds uniformly on compact sets if the function h in question is either (Lebesgue) , where we obtain sets of conditions on h, each necessary and sufficient for UCT (see Theorem UCT below). In [BOst2] this motivates a unified approach to the Karamata theory of the two cases by regarding each as a subfamily of a single family of functions, one that is defined by combinatorial properties shared by both. An alternative unification (see [BOst3]) derives the measure and category forms of their shared infinite combinatorics from a single topological result, the Category Embedding Theorem, by specialization to two topologies-the Euclidean topology and the density topology.A very few papers in regular variation are able to make progress without imposing regularity conditions. Foremost among these are the Ash-Erdős-Rubel paper [AER], where a growth condition is used instead, and the work of Heiberg [Hei]