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Uniform modular integrability and convergence properties for a class of Euryon integral operators in function space. (English) Zbl 1141.41006
The aim of the work is to generalize a modular convergence result, obtained by the authors themselves in a previous paper and concerning a class of Euryon operators in suitable Orlicz spaces of a locally compact Hausdorff topological group G. The generalization is twofold: on one hand, the algebraic structure of the underlying space G is removed, so the Orlicz spaces involved are built on an abstract locally compact Hausdorff space: this is the content of the first part of the paper, on the other hand, the structure of Orlicz space is then generalized to an abstract modular space, and the corresponding convergence of the operators is then replaced by the modular one: this is the concern of the second part of the paper. The key tool in order to achieve such generalizations is a notion of equi-integrability for the kernels involved, which replaces the condition of compact support. So, assuming that the family of kernels \((K_w)\) is singular, and that reasonable compatibility conditions with the modular involved are satisfied, the main theorem asserts that from the equi-integrability condition it follows modular convergence of the Urysohn operators \((T_w)\) related to the given kernels in an appropriate modular space on \(G\). Besides the interest of the results in themselves, and the clearness and correctness of the proof, a number of examples and explanations are to be mentioned, making the paper almost self-contained and complete.

MSC:
41A35 Approximation by operators (in particular, by integral operators)
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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