论文标题
用于广义采样的最短支持的多型碱基
Shortest-support Multi-Spline Bases for Generalized Sampling
论文作者
论文摘要
广义采样在于恢复功能$ f $的恢复，从线性换档系统集合的响应的样本到输入$ f $。重建的功能通常是有限生成的整数转移空间的成员，该空间可以将多项式繁殖至给定的$ M $。虽然此属性允许订单$（M+1）$的近似功率，但在基本功能的支持长度上取决于折衷。具体而言，我们证明发电机支持的长度至少为$（M+1）$。遵循此结果，我们介绍了$ m $的最短基础的概念，这是由于我们希望最大程度地降低计算成本的愿望。然后，我们证明，最短支持的任何基础都会产生Riesz的基础。最后，我们介绍了一种递归算法，以构建任何多键空间的最短支持基础。它提供了多项式和Hermite B-Splines的概括。该框架为新型应用铺平了道路，例如具有任意高近似功率的快速衍生化采样。
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.
