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. 
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     title = {Direct and inverse theorems of approximation theory for the generalized modulus of smoothness of some class of functions},
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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/

[1] M.K. Potapov, “Ob usloviyakh sovpadeniya nekotorykh klassov funktsii”, Trudy seminara im. I.G. Petrovskogo, 6, 1981, 223–238 | Zbl

[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.

[3] N.Ya. Vilenkin, Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1965

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