Smoothness theorems for generalized symmetric Pollaczek weights on (−1,1)

In this note a new characterization of smoothness is obtained for weighted polynomial approximation in Lp (1 ⩽ p ⩽ ∞) with respect to a large class of exponential weights in (−1,1) which include the classical Pollaczek weights. Along the way we prove Marchaud inequalities, saturation theorems, existence theorems for derivatives and generalize a theorem of D. Lubinsky.