  Variable Muckenhoupt weights are considered in variable exponent Lebesgue
spaces. Applications are given for polynomial approximation in these spaces.
Boundedness of averaging operator is proved to gain a transference result.
Almost all weighted norm inequalities are obtained using this transference
result. Potential type approximate identities are considered. Translations of
Steklov operator are proved to be bounded in these spaces. K-functional is a
good apparatus for measuring smoothness properties of functions given in these
function classes. Main inequalities of approximation are derived.
