Geodesic
Direct and inverse theorems of approximation theory for the generalized modulus of smoothness of some class of functions
Problemy fiziki, matematiki i tehniki, no. 4 (2021), pp. 92-94

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This article proves the coincidence of generalized moduli of smoothness of the $k$-s order, defined with the help of Gegenbauers generalized shift operator, with different and identical shifts and, as a consequence, direct and inverse theorems of approximation theory by algebraic polynomials are obtained.
Keywords: the best approximation by algebraic polynomials, Gegenbauers generalized shift operator, generalized modulus of smoothness. 
G. N. Kazimirov. Direct and inverse theorems of approximation theory for the generalized modulus of smoothness of some class of functions. Problemy fiziki, matematiki i tehniki, no. 4 (2021), pp. 92-94. http://geodesic.mathdoc.fr/item/PFMT_2021_4_a14/
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[2] G.N. Kazimirov, O teoremakh Dzheksona dlya $k$-go obobschennogo modulya gladkosti, Dep. v VINITI RAN 27.12.94, No 3054-V94, 40 pp.

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[4] G.N. Kazimirov, “O sovpadenii obobschennykh modulei gladkosti na nekotorom klasse funktsii”, Problemy fiziki, matematiki i tekhniki, 2020, no. 2 (43), 69–70