Geodesic
Equivalent structural characteristic of the given generalized modulus of smoothness
Problemy fiziki, matematiki i tehniki, no. 3 (2010), pp. 49-51.

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In this paper the equivalence of the $k$-th generalized modulus of smothness determined with the help of the generalized shift operator and Peetres $K$-functional is proved.
Keywords: Jacobi generalized shift operator, generalized modulus of smoothness, $K$-functional. 
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G. N. Kazimirov. Equivalent structural characteristic of the given generalized modulus of smoothness. Problemy fiziki, matematiki i tehniki, no. 3 (2010), pp. 49-51. http://geodesic.mathdoc.fr/item/PFMT_2010_3_a6/

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[2] M. K. Potapov, “O strukturnykh kharakteristikakh klassov funktsii s dannym poryadkom nailuchshego priblizheniya”, Trudy MIAN SSSR, 134, 1975, 260–277 | MR | Zbl

[3] G. N. Kazimirov, Priblizhenie algebraicheskimi mnogochlenami funktsii s dannym $k$-m obobschennym modulem gladkosti, Dis. ... kand. fiz.-mat. nauk: 01.01.01, M., 1995, 106 pp.