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.

翻译：本文研究平移不变空间中涉及导数的采样与插值问题，以及导数采样展开对广泛函数类的误差分析。针对重数 $r>1$ 的情形，引入了一类不同于欧拉-弗罗贝尼乌斯多项式的新多项式。利用洛朗算子给出了均匀采样与导数的完整刻画。对于由紧支撑函数生成的平移不变空间，以 $L^p$ 平均光滑模建立了信号（不必连续）通过导数采样展开的逼近速率。最后，详细讨论了若干典型算例以说明各类问题。