A Note on the Mellin-Fejer Kernels

In this paper, we present direct approximation results for Mellin-Fejer type operators. In particular, we are interested in quantitative approximation theorems in Mellin-Lebesgue spaces, using a suitable modulus of smoothness. Furthermore, an asymptotic formula of Voronovskaya type is obtained. Introducing a new modulus of smoothness, called logarithmic weighted modulus of smoothness in the weighted Mellin-Lebesgue spaces, a rate of convergence is obtained, and the global smoothness preservation property is also expressed through the new modulus of smoothness. The present results represent a completion of the theory developed in recent years on this subject. Finally, we illustrate some particular examples of kernel functions satisfying the assumptions of the theorems.