Moduli of Smoothness Related to Fractional Riesz-Derivatives

Abstract

New moduli of smoothness ${\omega }_{⟨\beta ⟩}(f,\delta {)}_p$, $0<\beta <1$, related to the Riesz derivative of order $\beta$ are introduced. Their properties are studied in $L_p$-spaces of $2\pi$-periodic functions for $0<p\le +\infty$. The modulus ${\omega }_{⟨\beta ⟩}(f,\delta {)}_p$ is shown to be equivalent to the polynomial $K$-functional associated with the corresponding Riesz derivative. As a consequence direct Jackson-type and inverse Bernstein-type estimates are proved, the quality of approximation by methods generated by Riesz kernels is described in terms of the these moduli.