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Approximation properties in abstract modular spaces for a class of general sampling-type operators. (English) Zbl 1089.41012
In recent years the authors have studied approximation properties for families of nonlinear integral operators of the form \[ (T_wf)(s)=\int_{H_w}K_w(s,t,f(t)){d\mu}_{H_w}(t),\quad s\in G,\; w>0, \tag{1} \] where \(G\) is a locally compact metric space, \((H_w)_{w>0}\) is a net of closed subsets of \(G\) such that \(\overline{U_{w>0}H_w}=G\) and, for every \(w>0\), \({\mu}_{H_w}\) is a regular measure on \(H_w\). For these operators the authors studied estimates and convergence in Orlicz spaces. Very interesting families of operators of discrete type can be obtained by specializing \(G\) and \(H_w\): generalized sampling series of a function (introduced by P.L.Butzer and his school), Kramer-type operators, Bernstein, Szász-Mirakjan, Baskakov operators, and so on. The aim of this paper is to study approximation properties for the operators (ref{1}) in abstract modular function spaces. Pointwise and uniform convergence is investigated for bounded continuous functions on \(G\). A modular convergence theorem is established for functions in suitable subspaces of modular spaces. The paper contains also some results about the regularity of the summability method induced by the family (ref{1}). The general results are explicitly discussed in several particular cases.

MSC:
41A35 Approximation by operators (in particular, by integral operators)
41A25 Rate of convergence, degree of approximation
47G10 Integral operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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