  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.
