Jackson-Stechkin type inequalities for differentiable functions in weighted Orlicz spaces

In the present work some Jackson Stechkin type direct theorems of trigonometric approximation are proved in Orlicz spaces with weights satisfying some Muckenhoupt's $A_{p}$ condition. To obtain refined version of the Jackson type inequality we prove an extrapolation theorem, Marcinkiewicz multiplier theorem and Littlewood Paley type results. As a consequence refined inverse Marchaud type inequalities are obtained. By means of a realization result we find an equivalence between the fractional order weighted modulus of smoothness and the classical weighted Peetre's K-functional.