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.

中文翻译：

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加权空间中整个插值的平均收敛性

我们研究了整个拉格朗日插值和指数类型\(\tau \)的 Hermite 插值的收敛性，如\(\tau \rightarrow \infty \)，在实线上的加权\(L^p\) -空间中。权重是依赖于\(\tau \)的整个函数的倒数，可以被视为目标权重w 的平滑版本。收敛性陈述是从整个函数的加权 Marcinkiewicz 不等式中获得的。我们应用我们的主要结果来处理功率权重。