Generalized sampling consists in the recovery of a function $f$, from the samples of the responses of a collection of linear shift-invariant systems to the input $f$. The reconstructed function is typically a member of a finitely generated integer-shift-invariant space that can reproduce polynomials up to a given degree $M$. While this property allows for an approximation power of order $(M+1)$, it comes with a tradeoff on the length of the support of the basis functions. Specifically, we prove that the sum of the length of the support of the generators is at least $(M+1)$. Following this result, we introduce the notion of shortest basis of degree $M$, which is motivated by our desire to minimize the computational costs. We then demonstrate that any basis of shortest support generates a Riesz basis. Finally, we introduce a recursive algorithm to construct the shortest-support basis for any multi-spline space. It provides a generalization of both polynomial and Hermite B-splines. This framework paves the way for novel applications such as fast derivative sampling with arbitrarily high approximation power.
翻译:普遍抽样包括从收集的线性变换系统的答复样本中回收一个功能f美元到投入f美元。重组功能通常是一个有限生成的整变变换空间的成员,可以复制多元值,但最高可复制到一定程度 $M美元。虽然该属性允许排序近似能量$(M+1),但在支持基础功能的长度上有一个权衡。具体地说,我们证明发电机支持的长度总和至少为$(M+1)美元。在此结果之后,我们引入了最短的M美元基数概念,其动机是我们希望最大限度地减少计算成本。然后我们证明,任何最短支持基础都会产生Riesz基数。最后,我们引入一种循环算法,为任何多样空间构建最短的支持基础。它提供了多色和赫米特B样的概括性。这个框架铺平了新应用的途径,例如任意高压的快速衍生物取样。