Universal sampling discretization

Feng Dai; V. Temlyakov

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arXiv · 2107.11476

Universal sampling discretization

Feng Dai·V. Temlyakov

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Abstract

Let $X_N$ be an $N$-dimensional subspace of $L_2$ functions on a probability space $(\Omega, \mu)$ spanned by a uniformly bounded Riesz basis $\Phi_N$. Given an integer $1\leq v\leq N$ and an exponent $1\leq q\leq 2$, we obtain universal discretization for integral norms $L_q(\Omega,\mu)$ of functions from the collection of all subspaces of $X_N$ spanned by $v$ elements of $\Phi_N$ with the number $m$ of required points satisfying $m\ll v(\log N)^2(\log v)^2$. This last bound on $m$ is much better than previously known bounds which are quadratic in $v$. Our proof uses a conditional theorem on universal sampling discretization, and an inequality of entropy numbers in terms of greedy approximation with respect to dictionaries.

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BibTeX RIS

Feng Dai, V. Temlyakov. 2021-07-23. Universal sampling discretization. https://arxiv.org/abs/2107.11476

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