Sequences of composition operators in spaces of functions of bounded Φ-variation

Abstract

The main results of the paper are contained in Theorems 1 and 2. Theorem 1 presents necessary and sufficient conditions for a sequence of functions hn: 〈c, d〉 → 〈a, b〉, n = 1, 2, ..., to have bounded sequences of Ψ-variations {VΨ (〈c, d〉; f ◦ hn)}n=1∞ evaluated for the compositions of an arbitrary function f: 〈a, b〉 → ℝ with finite Φ-variation and the functions hn. In Theorem 2, the same is done for a sequence of functions hn: ℝ → ℝ, n = 1, 2, ..., and the sequence of Ψ-variations {VΨ(〈a, b〉; hn ◦ f)}n=1∞.