Moduli of Smoothness Related to Fractional Riesz-Derivatives

  • Konstantin V. Runovski

    Lomonosov State University, Sevastopol, Ukraine
  • Hans-Jürgen Schmeisser

    Friedrich-Schiller-University, Jena, Germany

Abstract

New moduli of smoothness , , related to the Riesz derivative of order are introduced. Their properties are studied in -spaces of -periodic functions for . The modulus is shown to be equivalent to the polynomial -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.

Cite this article

Konstantin V. Runovski, Hans-Jürgen Schmeisser, Moduli of Smoothness Related to Fractional Riesz-Derivatives. Z. Anal. Anwend. 34 (2015), no. 1, pp. 109–125

DOI 10.4171/ZAA/1531