使用自适应字典的非线性逼近
Nonlinear approximation with adaptive dictionaries
AI总结：
研究关于自适应字典的非线性逼近问题，通过发展一种一般方法，研究由特定积分算子定义的类集合的采样恢复误差渐近行为，此前该方法已用于柯尔莫哥洛夫宽度和熵数。
AI中文摘要：
众所周知，对函数类的柯尔莫哥洛夫宽度的研究，即具有核\(K\)的积分算子\(J_K\)的\(L_q\)空间单位球的像，与关于经典双线性字典的核\(K\)的稀疏逼近研究密切相关。最近发现，如果研究相同类的最优线性采样恢复误差，而不是柯尔莫哥洛夫宽度，那么需要研究关于由核\(K\)确定的自适应字典的核\(K\)的稀疏逼近。本文研究了关于自适应字典的非线性逼近这一重要问题。此外，本文继续发展了一种与上述非线性逼近问题相关的一般方法。我们研究采样恢复误差的渐近行为，不是针对单个光滑度类，而是针对由具有来自给定函数类的核的积分算子定义的类的集合。之前这种方法已用于柯尔莫哥洛夫宽度，最近也用于熵数。
英文摘要：
It is well known that the study of the Kolmogorov widths of a function class, which is the image of the unit ball of the $L_q$ space of an integral operator $J_K$ with the kernel $K$, is closely connected with the study of sparse approximations of the kernel $K$ with respect to the classical bilinear dictionary. Recently, it was discovered that if instead of the Kolmogorov widths we study the errors of optimal linear sampling recovery of the same classes, then we need to study sparse approximations of the kernel $K$ with respect to an adaptive dictionary, which is determined by the kernel $K$. In this paper we study this important problem of nonlinear approximation with respect to an adaptive dictionary. Also, in this paper we continue to develop the following general approach, which is related to the above nonlinear approximation problem. We study asymptotic behavior of the errors of sampling recovery not for an individual smoothness class, how it is usually done, but for the collection of classes, which are defined by integral operators with kernels coming from a given class of functions. Earlier, such approach was realized for the Kolmogorov widths and very recently for the entropy numbers.
