Geodesic
On the Connection Between the Best Approximation by Algebraic Polynomials and the Modulus of Smoothness of Order~$r$
Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 149-164

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We introduce a family of asymmetric generalized translations. In terms of these operators we define generalized miduli of smoothness for which the direct and inverse theorems of approximation theory are proved. 
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     author = {M. K. Potapov and F. M. Berisha},
     title = {On the {Connection} {Between} the {Best} {Approximation} by {Algebraic} {Polynomials} and the {Modulus} of {Smoothness} of {Order~}$r$},
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M. K. Potapov; F. M. Berisha. On the Connection Between the Best Approximation by Algebraic Polynomials and the Modulus of Smoothness of Order~$r$. Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 149-164. http://geodesic.mathdoc.fr/item/CMFD_2007_25_a11/