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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 $$L^0(G)$$.

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