Widths and rigidity
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910310248480768 |
|---|---|
| author | Malykhin, Yuri |
| author_facet | Malykhin, Yuri |
| contents | 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 $L_p$, $p<1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_03453 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Widths and rigidity Malykhin, Yuri Functional Analysis 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 $L_p$, $p<1$. |
| title | Widths and rigidity |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2205.03453 |