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Mean convergence of entire interpolations in weighted space. (English) Zbl 1471.41003

Summary: We investigate the convergence of entire Lagrange interpolations and of Hermite interpolations of exponential type \(\tau \), as \(\tau \rightarrow \infty \), in weighted \(L^p\)-spaces on the real line. The weights are reciprocals of entire functions that depend on \(\tau\) 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.

MSC:

41A05 Interpolation in approximation theory
30E05 Moment problems and interpolation problems in the complex plane
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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