有界域上的采样定理
Sampling theorems on bounded domain
AI总结：
本文提出了一种从非均匀采样中完美重构有界流形上多分辨率空间函数的最优迭代方案，并给出了误差分析及数值例子。
摘要：
本文关注从非均匀采样中完美重构属于有界流形上多分辨率空间的函数的迭代方案。这些方案具有最优复杂度，即达到特定固定精度的计算成本与计算量成正比。由于迭代一致收敛，可以产生相应的迭代积分方案，从而从非均匀采样中恢复属于多分辨率空间的函数的积分。我们还给出了误差分析，特别地，我们估计了从非均匀采样中恢复光滑函数（属于$H^s$，但不一定属于任何多分辨率空间）及其积分时产生的$L^2$误差。展示并讨论了多个单变量和双变量数值例子。我们还表明，可以从实线上由所谓GP函数形成的某些可细化基构造流形上相当多样化的多分辨率空间。这类函数（特别包含B样条）在产生良态基方面具有显著性质。所得的多分辨率分析非常适合应用从非均匀采样中迭代恢复函数的方法。
英文摘要：
This paper concerns with iterative schemes for the perfect reconstruction of functions belonging to multiresolution spaces on bounded manifolds from nonuniform sampling. The schemes have optimal complexity in the sense that the computational cost to achieve a certain fixed accuracy is proportional to the computed quantity. Since the iterations converge uniformly, one can produce corresponding iterative integration schemes that allow to recover the integral of functions belonging to multiresolution spaces from nonuniform sampling. We present also an error analysis and, in particular, we estimate the $L^2$-error which one produces in recovering smooth functions in $H^s$, but not necessarily in any multiresolution space, and their integrals from nonuniform sampling. Several uni- and bi-variate numerical examples are illustrated and discussed. We also show that one can construct a rather large variety of multiresolution spaces on manifolds from certain refinable bases on the real line formed by so-called GP-functions. This class of functions that contains in particular B-splines has remarkable properties in terms of producing well-conditioned bases. The resulting multiresolution analyses are well-suited for the application of the iterative recovering of functions from nonuniform sampling.
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