The analogs of the Korovkin theorems in banach function spaces


ZEREN Y., Ismailov M., Karacam C.

POSITIVITY, vol.26, no.2, 2022 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 26 Issue: 2
  • Publication Date: 2022
  • Doi Number: 10.1007/s11117-022-00897-y
  • Journal Name: POSITIVITY
  • Journal Indexes: Science Citation Index Expanded, Scopus, ABI/INFORM, Business Source Elite, Business Source Premier, MathSciNet, zbMATH
  • Keywords: Korovkin theorems, Banach function spaces, Boyd indices, Shift operator, Kantorovich polynomial, Hardy-Littlewood maximal operator, PIECEWISE-LINEAR PHASE, MORREY SPACES, SYSTEM, EXPONENTS, OPERATORS, LEBESGUE, BASICITY

Abstract

This work is dedicated to Korovkin type theorems in Banach function spaces. The subspace X-S of the Banach function space X generated by the shift operator is considered and the density of the set C-0(infinity) in X-S is proved. The analogs of the Korovkin theorems in X-S are obtained. Also, the analog of the Korovkin theorem for Kantorovich polynomials is derived both in the cases of rearrangement-invariant and general non-rearrangement-invariant Banach function spaces. These results are obtained for Lebesgue spaces, grand-Lebesgue spaces, Morrey-type spaces and their weighted versions, weak Lebesgue spaces, Orlicz spaces. Note that in our case the Korovkin-type theorem for Kantorovich polynomials in Morrey spaces is an only natural analog of the classical L-p version of the Korovkin theorem and strengthens the previously known result.