Mean Convergence of Entire Interpolations in Weighted Space

Abstract

We investigate the convergence of entire Lagrange interpolations and of Hermite interpolations of exponential type $$\tau $$ τ , as $$\tau \rightarrow \infty $$ τ → ∞ , in weighted $$L^p$$ L p -spaces on the real line. The weights are reciprocals of entire functions that depend on $$\tau $$ τ and may be viewed as smoothed versions of a target weight w. The convergence statements are obtained from weighted Marcinkiewicz inequalities for entire functions. We apply our main results to deal with power weights.