信号在$L^p({\Bbb R}^d)$再生核子空间中的采样与重构
Sampling and Reconstruction of Signals in a Reproducing Kernel Subspace of $L^p({\Bbb R}^d)$
摘要：
本文研究了在$L^p(\Rd), 1\le p\le \infty$的再生核子空间中信号的采样与重构问题，该再生核子空间中的积分算子具有非对角衰减性和正则性. $p$ -可积非均匀样条空间和由多个局部化函数生成的平移不变空间是$L^p(\Rd)$再生核子空间的典型例子。我们表明，在这样的再生核子空间中的信号可以重建在一个稳定的方式，从它的样本上采取一个相对分离的集足够小的差距。我们还研究了指数收敛性，一致性，和渐近逐点误差估计的迭代近似投影算法和迭代框架算法重建的信号在这些再生核空间从它的样本具有足够小的差距。
英文摘要：
In this paper, we consider sampling and reconstruction of signals in a reproducing kernel subspace of $L^p(\Rd), 1\le p\le \infty$, associated with an idempotent integral operator whose kernel has certain off-diagonal decay and regularity. The space of $p$-integrable non-uniform splines and the shift-invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of $L^p(\Rd)$. We show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively-separated set with sufficiently small gap. We also study the exponential convergence, consistency, and the asymptotic pointwise error estimate of the iterative approximation-projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap.
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