包含覆盖不连续信号误差估计的平移不变空间中的导数采样展开
Derivative sampling expansions in shift-invariant spaces with error estimates covering discontinuous signals
AI总结：
本文研究平移不变空间中涉及导数的采样与插值问题，引入基于导数样本的新型多项式，利用 Laurent 算子刻画均匀采样，并基于 $L^p$-平均光滑模建立了对不连续信号的逼近速率。
摘要：
本文研究平移不变空间中涉及导数的采样与插值问题，以及对基本大类函数的导数采样展开的误差分析。引入了一种基于导数样本的新型多项式，它不同于重数 $r>1$ 的 Euler-Frobenius 多项式。利用 Laurent 算子给出了带导数的均匀采样的完整刻画。由紧支撑函数生成的平移不变空间中，导数采样展开对信号（不要求连续）的逼近速率通过 $L^p$-平均光滑模得以建立。最后，详细讨论了说明各类问题的几个典型实例。
英文摘要：
This paper is concerned with the problem of sampling and interpolation involving derivatives in shift-invariant spaces and the error analysis of the derivative sampling expansions for fundamentally large classes of functions. A new type of polynomials based on derivative samples is introduced, which is different from the Euler-Frobenius polynomials for the multiplicity $r>1$. A complete characterization of uniform sampling with derivatives is given using Laurent operators. The rate of approximation of a signal (not necessarily continuous) by the derivative sampling expansions in shift-invariant spaces generated by compactly supported functions is established in terms of $L^p$- average modulus of smoothness. Finally, several typical examples illustrating the various problems are discussed in detail.
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