Riesz spaces with generalized Orlicz growth

Abstract

We consider a Riesz ϕ-variation for functions f defined on the real line when ϕ : Ω × [0, ∞) → [0, ∞) is a generalized Φ-function. We show that it generates a quasi-Banach space and derive an explicit formula for the modular when the function f has bounded variation. The resulting BV -type energy has previously appeared in image restoration models. We generalize and improve previous results in the variable exponent and Orlicz cases and answer a question regarding the Riesz-Medvedev variation by Appell, Banaś and Merentes [Bounded Variation and Around,