Separability of the space of continuous functions that are continuous in Λ-variation

The Shao–Sablin index of a Λ-sequence Λ=(λi) is defined by . The main result of the paper states that the Banach space CΛBV of continuous functions of bounded Λ-variation with the standard Λ-variation norm is separable if and only if SΛ<2. Also, ΛBV=ΛBVc if and only if SΛ<2, where ΛBVc denotes the space of functions continuous in Λ-variation. A number of corollaries is drawn, and one of them being that the Garsia–Sawyer class GS is a dense subset of the Banach space HBV of functions of bounded harmonic variation.