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On approximation properties of Urysohn integral operators. (English) Zbl 1012.41017
Let \(G\) be a locally compact Hausdorff topological space provided with a Borel measure \(\mu\). There are studied problems of approximation of a function \(f\) by means of a family of operators \[ (T_wf)(s)= \int_GK_w \bigl (s,t,f(t)\bigr) d\mu(t),\quad w<0,\;s\in G \] as \(w\to+\infty\), both in the sense of uniform convergence and in the sense of modular convergence generated by means of a modular \(\rho\) defined in the space \(L^0(G)\) of measurable, finite \(\mu\)-a.e. functions \(f:G\to \mathbb{R}\). Applying singularity assumptions on the family of kernel functions \(K:G\times G\times \mathbb{R}\to\mathbb{R}\) and suitable assumptions on the modulars \(\rho,\eta\) in \(L^0(G)\), there are obtained approximation results of the form \(T_wf\to f\) pointwise, \(\|T_wf-f\|_{ C(G)} \to 0\) as \(w\to+ \infty\) for \(f\in C(G)\), and \(\rho(\lambda (T_wf-f))\to 0\) as \(w\to+ \infty\) for some \(\lambda>0\) (dependent on \(f)\) for functions \(f\) belonging to a modular space generated by the sum of modulars \(\rho+ \eta\) in \(L^0(G)\).

41A25 Rate of convergence, degree of approximation
41A35 Approximation by operators (in particular, by integral operators)
47A58 Linear operator approximation theory