On some constructions of a non-periodic modulus of smoothness related to the Riesz derivative

A new non-periodic modulus of smoothness related to the Riesz derivative is constructed. Its properties are studied in the spaces $L_p(\mathbb{R})$ of non-periodic functions with $1\leqslant p\leqslant+\infty$. The direct Jackson type estimate is proved. It is shown that the introduced modulus is equivalent to the $K$-functional related to the Riesz derivative and to the approximation error of the convolution integrals generated by the Fejér kernel.