Direct and inverse theorems of approximation theory for the \(m\)th generalized modulus of smoothness (Q2783083)

The author introduces a generalised modulus of continuity replacing the standard shift operator by NEWLINE\[NEWLINETy(f,x,r,s): ={2^{s+r} \over\pi} \int^{+1}_{-1} f(R)\psi (x,y,z){dz \over \sqrt{1-z^2}},NEWLINE\]NEWLINE where \(R:=xy+z \sqrt{1-x^2}\sqrt {1-y^2}\) and \(\psi\) be defined as a complicated rational function of \(\sqrt {1-x^2}\), \(\sqrt {1-y^2}\) and \(\sqrt {1-R^2}\). Using these moduli the author proves direct and inverse theorems of the Bernstein-Jackson type for approximation of algebraic polynomials in \(L_p(-1,1)\).NEWLINENEWLINEFor the entire collection see [Zbl 0981.00017].