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Article

Approximation by families of generalized sampling series, realizations of generalized K-functionals and generalized moduli of smoothness

Journal of Mathematical Analysis and Applications. 2020. Vol. 489. No. 1. P. 1-19.
Artamonov S., Runovski K., Schmeisser H.

We study properties of realizations of generalized K-functionals and generalized moduli of smoothness in Lp(Rd) spaces with 0<p<∞ as well as in the space C(Rd) of uniformly continuous and bounded functions. We obtain direct Jackson-type estimates and inverse Bernstein-type estimates. We show the equivalence between the approximation error of families of generalized sampling series generated by an arbitrary generator with compact support, realizations of generalized K-functionals generated by homogeneous functions and generalized moduli of smoothness generated by a periodic function. Our approach covers classical approximation methods, K-functionals related to fractional derivatives and fractional moduli of smoothness.