Geodesic
Direct and Inverse Theorems of Approximation Theory for the $m$th Generalized Modulus of Smoothness
Informatics and Automation, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 289-297

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The relation between polynomial approximations of functions and their generalized $m$th moduli of smoothness defined with the use of a two-parameter family of nonsymmetric generalized translation operators is examined. 
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     author = {M. K. Potapov},
     title = {Direct and {Inverse} {Theorems} of {Approximation} {Theory} for the $m$th {Generalized} {Modulus} of {Smoothness}},
     journal = {Informatics and Automation},
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     volume = {232},
     year = {2001},
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     url = {http://geodesic.mathdoc.fr/item/TRSPY_2001_232_a23/}
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M. K. Potapov. Direct and Inverse Theorems of Approximation Theory for the $m$th Generalized Modulus of Smoothness. Informatics and Automation, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 289-297. http://geodesic.mathdoc.fr/item/TRSPY_2001_232_a23/