This short chapter may be viewed as a complement to the chapters on almost periodicity. Its goal is a fairly self-contained account of some averaging processes of functions along sequences of the form, where α is a fixed real number with and is arbitrary. Such sequences appear in the spectral theory of inflation systems in various ways. Due to the connection with uniform distribution theory, the results will mostly be metric in nature, which means that they hold for Lebesgue-almost every.
A frequently encountered problem in mathematics and its applications is the study of averages of the form, where f is a function with values in C or, more generally, in some Banach space, and is a sequence of numbers in the domain of f. Quite often, an exact treatment of these averages is out of hand, and one resorts to the analysis of asymptotic properties for large N. This, for instance, is common in analytic number theory; compare [18, 19, 1] and references therein. Equally important is the case where one can establish the existence of a limit as N → ∞, and then calculate it. This occupies a good deal of ergodic theory, where Birkhoff's theorem and Kingman's subadditive theorem provide powerful tools to tackle the problem; see [14, 37] for background.
However, not all tractable cases present themselves in a way that is immediately accessible to tools from ergodic theory. Also, depending on the nature of the underlying problem, one might prefer a more elementary method, as Birkhoff-type theorems already represent a fairly advanced kind of ‘weaponry’. An interesting (and certainly not completely independent) approach is provided by the theory of uniform distribution of sequences, which essentially goes back to Weyl  and has emerged as a major tool for the study of function averages, in particular for functions that are periodic or defined on a compact domain; see [24, 16, 25] and references therein for more.