Results on two different settings of asymptotic behavior of approximation characteristics of individual functions are presented. First, we discuss the following classical question for sparse approximation. Is it true that for any individual function from a given function class its sequence of errors of best sparse approximations with respect to a given dictionary decays faster than the corresponding supremum over the function class? Second, we discuss sparse approximation by greedy type algorithms. We show that for any individual function from a given class we can improve the upper bound on the rate of convergence of the error of approximation by a greedy algorithm if we use some information from the previous iterations of the algorithm. Bounds of this type fall in the category of a posteriori bounds.

给出了在两种不同的渐近行为设置下，各个函数的逼近特征的结果。首先，我们讨论以下关于稀疏近似的经典问题。对于来自给定函数类的任何单个函数，其相对于给定字典的最佳稀疏近似误差序列的衰减速度是否快于该函数类的相应最高阈值的衰减速度？其次，我们讨论贪婪类型算法的稀疏近似。我们表明，对于给定类的任何单个函数，如果我们使用算法先前迭代中的某些信息，则可以通过贪婪算法提高近似误差收敛速度的上限。这种类型的边界属于后验边界的类别。