论文标题
宽度和刚性
Widths and rigidity
论文作者
论文摘要
我们认为有限函数集的Kolmogorov宽度。 $ n $函数的任何正顺式系统都在$ l_2 $中刚性，即，尺寸基本小于$ n $的线性子空间不能很好地近似。对于较弱的指标而言，情况并非如此：众所周知，在每个$ l_p $，$ p <2 $中，第一个$ n $ walsh函数可以是$ o（1）$ - 由尺寸$ o（n）$的线性空间近似。我们为刚性提供一些足够的条件。我们证明，功能的独立性（以概率含义）意味着$ l_1 $甚至$ l_0 $的刚度 - 与衡量收敛相对应的度量。在$ L_P $，$ 1 <P <2 $的情况下，条件较弱：任何$ S_ {P'} $ - 系统为$ L_P $ -RIGID。同样，我们获得了一些积极的结果，例如第一个$ n $三角函数可以通过$ l_0 $的非常低维空间近似，并且由$ o（n）$谐波生成的子空间中的子空间，$ l_p $，$ p <1 $。
We consider Kolmogorov widths of finite sets of functions. Any orthonormal system of $N$ functions is rigid in $L_2$, i.e. it cannot be well approximated by linear subspaces of dimension essentially smaller than $N$. This is not true for weaker metrics: it is known that in every $L_p$, $p<2$, the first $N$ Walsh functions can be $o(1)$-approximated by a linear space of dimension $o(N)$. We give some sufficient conditions for rigidity. We prove that independence of functions (in the probabilistic meaning) implies rigidity in $L_1$ and even in $L_0$ -- the metric that corresponds to convergence in measure. In the case of $L_p$, $1<p<2$, the condition is weaker: any $S_{p'}$-system is $L_p$-rigid. Also we obtain some positive results, e.g. that first $N$ trigonometric functions can be approximated by very-low-dimensional spaces in $L_0$, and by subspaces generated by $o(N)$ harmonics in $L_p$, $p<1$.
