Density Theorems for Nonuniform Sampling of Bandlimited Functions Using Derivatives or Bunched Measurements
Density Theorems for Nonuniform Sampling of Bandlimited Functions Using Derivatives or Bunched Measurements
复制标题
DOI：
10.1007/s00041-016-9504-8
复制
发表时间：
2017-12-01
影响因子：
1.2
通讯作者：
Hansen, Anders C.
中科院分区：
文献类型：
--
作者：
Adcock, Ben;Gataric, Milana;Hansen, Anders C.
我们提供了一组非均匀样本的充分密度条件，以产生一组多变量带限函数的采样时，测量组成的逐点评估的功能和它的第一个k衍生物。沿着与相应的框架边界的明确估计，我们得到明确的密度界，并表明，随着k的增加，它的线性增长与比例常数。寻求更大的间隙条件下，我们还证明了一个多元扰动结果的非均匀样本是足够接近的采样集，例如，均匀的采样时间的奈奎斯特速率。此外，在单变量设置中，我们考虑所谓的非均匀聚束采样的相关问题，其中在每个采样间隔聚束测量的功能，并允许采样间隔是不同的长度。我们推导出一个明确的密度条件，线性增长的大s，与常数的比例依赖于束的宽度。聚束的宽度可以任意小，而且，对于足够窄的聚束和足够大的s，我们得到了与具有s导数的单变量采样的情况相同的结果。
We provide sufficient density condition for a set of nonuniform samples to give rise to a set of sampling for multivariate bandlimited functions when the measurements consist of pointwise evaluations of a function and its first k derivatives. Along with explicit estimates of corresponding frame bounds, we derive the explicit density bound and show that, as k increases, it grows linearly in with the constant of proportionality . Seeking larger gap conditions, we also prove a multivariate perturbation result for nonuniform samples that are sufficiently close to sets of sampling, e.g. to uniform samples taken at times the Nyquist rate. Additionally, in the univariate setting, we consider a related problem of so-called nonuniform bunched sampling, where in each sampling interval bunched measurements of a function are taken and the sampling intervals are permitted to be of different length. We derive an explicit density condition which grows linearly in for large s, with the constant of proportionality depending on the width of the bunches. The width of the bunches is allowed to be arbitrarily small, and moreover, for sufficiently narrow bunches and sufficiently large s, we obtain the same result as in the case of univariate sampling with s derivatives.