In previous papers, we used abstract potential theory, as developed by Puglede and Ohtsuka, to a systematic treatment of rendezvous numbers. We considered Chebyshev constants and energies as two variable set functions, and introduced a modified notion of rendezvous intervals which proved to be rather nicely behaved even for only lower semicontinuous kernels or for not necessarily compact metric spaces.Here we study the rendezvous and average numbers of possibly infinite dimensional normed spaces. It turns out that very general existence and uniqueness results hold for the modified rendezvous numbers in all Banach spaces. We also observe the connections of these magical numbers to Chebyshev constants, Chebyshev radius and entropy. Applying the developed notions with the available methods we calculate the rendezvous numbers or rendezvous intervals of certain concrete Banach spaces. In particular, a satisfactory description of the case of L p spaces is obtained for all p > 0.
Abstract. Consider a 1 , . . . , a n ∈ R arbitrary elements. We characterize those functions f : R → R that decompose into the sum of a j -periodic functions, i.e., f = f 1 +· · ·+f n with ∆ a j f (x) := f (x+a j )−f (x) = 0. We show that f has such a decomposition if and only if for all partitions B 1 ∪B 2 ∪· · ·∪B N = {a 1 , . . . , a n } with B j consisting of commensurable elements with least common multiplesActually, we prove a more general result for periodic decompositions of functions f : A → R defined on an Abelian group A; in fact, we even consider invariant decompositions of functions f : A → R with respect to commuting, invertible self-mappings of some abstract set A.We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.
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