Mean convergence of entire interpolations in weighted space

We investigate the convergence of entire Lagrange interpolations and of Hermite interpolations of exponential type in weighted $L^p$-spaces on the real line. The weights are reciprocals of entire functions and depend on the type and may be viewed as smoothed versions of a target weight. The convergence statements are obtained from Marcinkiewicz inequalities with constants proportional to the type. For the special case of power weights we recover results of Rahman and Grozev and of Lubinsky.