Geodesic
On some constructions of a non-periodic modulus of smoothness related to the Riesz derivative
Eurasian mathematical journal, Tome 9 (2018) no. 2, pp. 11-21.

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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. 
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S. Yu. Artamonov. On some constructions of a non-periodic modulus of smoothness related to the Riesz derivative. Eurasian mathematical journal, Tome 9 (2018) no. 2, pp. 11-21. http://geodesic.mathdoc.fr/item/EMJ_2018_9_2_a2/

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