### Abstract:

One of the most important problems in Approximation Theory is to connect the rate with which a function can be approximated and the smoothness of this function. The goal is to show direct and inverse estimates in terms of some measure of smoothness.
Typically, results are of the following type: ``a function can be approximated with a given order if and only if it belongs to a certain smoothness class".
We focus on the case of the weighted $\mathbb{L}_p[-1,1]$ spaces with not rapidly changing bounded not vanishing inside interval $(-1,1)$ weights. In order to describe
certain smoothness classes we will use moduli of smoothness $\omega^{ k}_{\phi}$ and $\omega^{\star k}_{\phi}$ and prove their equivalence.
As a final result, we will prove direct theorems for monotone and convex approximation.