Let G be a locally compact Hausdorff group with Haar measure, and let L⁰(G) be the space of extended real-valued measurable functions on G, finite a.e. Let ϱ and η be modulars on L⁰(G). The error of approximation ϱ(a(Tf - f)) of a function $f\in {\left(L⁰\left(G\right)\right)}_{\varrho +\eta }\cap DomT$ is estimated, where $\left(Tf\right)\left(s\right)={\int }_{G}K(t-s,f\left(t\right))dt$ and K satisfies a generalized Lipschitz condition with respect to the second variable.

Let X denote the space of all real, bounded double sequences, and let Φ, φ, Γ be φ-functions. Moreover, let Ψ be an increasing, continuous function for u ≥ 0 such that Ψ(0) = 0.In this paper we consider some spaces of double sequences provided with two-modular structure given by generalized variations and the translation operator (...).

We obtain modular convergence theorems in modular spaces for nets of operators of the form $\left(T_{w}f\right)\left(s\right)={\int }_H{K}_w(s-h_w\left(t\right),f\left(h_w\left(t\right)\right))d{\mu }_H\left(t\right)$, w > 0, s ∈ G, where G and H are topological groups and ${h_w}_{w>0}$ is a family of homeomorphisms $h_w:H\to h_w\left(H\right)\subset G.$ Such operators contain, in particular, a nonlinear version of the generalized sampling operators, which have many applications in the theory of signal processing.