Monotonicities of quasi-normed Calderón–Lozanovskiĭ spaces with applications to approximation problems
[ 1 ] Instytut Matematyki, Wydział Automatyki, Robotyki i Elektrotechniki, Politechnika Poznańska | [ P ] employee
2024
Journal year: 2024 | Journal volume: early view/in press
scientific article
english
- best dominated approximation
- monotonicity properties
- quasi-normed Calderón–Lozanovskiĭ space
- quasi-normed ideal space
- quasi-normed Orlicz spaces
EN We consider the geometric structure of quasi-normed Calderón–Lozanovskiĭ spaces. First, we study relations between the quasi-norm and the quasi-modular “near zero” and “near one,” which are fundamental for the theory. With their help, we provide a precise description of the basic monotonicity properties. In comparison with the well-known normed case, we develop a number of new techniques and methods, among which the conditions Δ𝜀 and Δ2−𝑠𝑡𝑟 play a crucial role. From our general results, we conclude the criteria for monotonicity properties in quasi-normed Orlicz spaces, which are new even in this particular context.We consider both the function and the sequence case aswell aswe admit degenerated Orlicz functions, which provides us with a maximal class of spaces under consideration. We also discuss the applications of suitable properties to the best dominated approximation problems in quasi-Banach lattices.
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