Riesz spaces with generalized Orlicz growth

We consider a Riesz $ϕ$-variation for functions $f$ defined on the real line when $φ:Ω\times[0,\infty)\to[0,\infty)$ 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 [\emph{Bounded Variation and Around}, Studies in Nonlinear Analysis and Applications, Vol. 17, De Gruyter, Berlin/Boston, 2014].

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