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In this paper we investigate the gamma-relative differentiation by themotivation of amending the order of the weighted polynomial approximation onthe semiaxis for certain functions. With the help of this we give somedefinitions of generalized Sobolev spaces, K-functionals and moduli ofsmoothness. We prove theorems for estimating these things with each other, inthe case of first order we prove equivalence. We remark some possibleapplications and other generalizations too.

Weighted gamma-K-functional and gamma-Modulus of Smoothness on the Semiaxis