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Modular approximation by a filtered family of sublinear operators. (English) Zbl 0636.46025

Let (\(\Omega\),\(\Sigma\),\(\mu)\) be a \(\sigma\)-finite measure space, let \({\mathcal X}\) be the space of extended real-valued \(\Sigma\)-measurable \(\mu\)-a.e. measurable functions on \(\Omega\), and let \(\rho\) : \(\Omega\times {\mathcal X}\to (0,\infty)\) be a modular function which is measurable with respect to the variable in \(\Omega\) for each function in \({\mathcal X}\) and is such that for x,y in \({\mathcal X}\), \(| x(s)| \leq | y(s)|\) implies that \(\rho\) (t,x)\(\leq \rho (t,y)\) for \(\mu\)- almost all t in \(\Omega\). Then, X, \(X_{\rho_ S}\) and \(\rho_ S\) are defined by
(i) \(\rho_ S(x)=\int_{\Omega}\rho (t,x)d\mu (t);\)
(ii) \(X=\{x\in {\mathcal X}\), \(\rho\) (t,ax)\(\to 0\) a.e. on \(\Omega\) as \(a\to 0\};\)
(iii) \(X_{\rho_ S}=\{x\in X:\rho_ S(ax)\to 0\) as \(a\to 0\}.\)
If V is an abstract set, \({\mathcal V}\) is a filter of subsets of V, \(g: V\to R\), then g(v)\(\to 0\) if for every \(\epsilon >0\), there is a set \(V_ 0\in {\mathcal V}\) such that \(| g(v)| <\epsilon\) for \(v\in V_ 0.\)
The main results of this paper relate to classes of q-sublinear or sublinear operators of the form \(\{T_ v\), \(v\in V\}\) which are \({\mathcal V}\)-bounded so that there are positive constants \(k_ 1,k_ 2\) and a function \(g: V\to R_+\), g(v)\(\to 0\), and for x,y in \(X_{\rho_ S}\) there is a set \(V_{x,y}\in {\mathcal V}\) such that \[ \rho_ S(a(T_ vx- T_ vy))\leq k_ 1\rho_ S(ak_ 2(x-y))+g(v),\quad for\quad v\in V_{x,y},\quad a>0. \] It is indicated that if \(\nu =0\) when operators are q-sublinear and \(\nu =-1\) when operators are sublinear, \(S_ 0\subseteq X_{\rho_ S}\), S is the set of finite linear combinations of elements of \(S_ 0\), with closure \(\bar S,\) and if \(\rho_ S(a(T_ vx+\nu x))\to 0\) for \(x\in S_ 0\), \(a>0\), then for \(x\in \bar S\) there is a number \(b>0\) such that \(\rho_ S(b(T_ vx+\nu x))\to 0\). Applications relating to approximation theory are stated.
Reviewer: G.O.Okikiolu

MSC:

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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