Approximation By Bandlimited Functions, Generalized K-Functionals and Generalized Moduli of Smoothness

We study properties of generalized K -functionals and generalized moduli of smoothness in L p (ℝ) spaces with 1 ≤ p < ∞ as well as in the space C (ℝ) of uniformly continuous and bounded functions. We obtain direct Jackson-type estimates and inverse Bernstein-type estimates. We show the equivalence between approximation error of convolution integrals generated by an arbitrary generator with compact support, generalized K -functionals generated by homogeneous functions and generalized moduli of smoothness. Our approach covers classical approximation methods, K -functionals related to fractional derivatives and fractional moduli of smoothness.