"Random sampling in a reproducing kernel subspace of $L^{p}({\mathbb R}^{n})$"Patel, DhirajThe random sampling problem in a higher dimensional Euclidean space was initially studied by Bass and Gr\"{o}chenig for the space of trigonometric polynomial and band-limited function. In the last decade, the problem has been intensively studied for the shift-invariant space and image space of an idempotent integral operator on $L^p({\mathbb R}^n)$. In this talk, we generalize the result for the localized reproducing kernel subspace of $L^p({\mathbb R}^n)$. We are concerned with the stability of the sample set uniformly distributed over a compact set $\Omega$ for the class of functions concentrated on $\Omega$. We show that if the sampling set on $\Omega$ discretizes the integral norm of simple functions up to a given error, then the sampling set is stable for the set of functions concentrated on $\Omega$. Moreover, we prove with an overwhelming probability that ${\mathcal O}(\mu(\Omega)(\log \mu(\Omega))^3)$ many random points uniformly distributed over $\Omega$ yield a stable set of sampling for functions concentrated on $\Omega$. This talk is based on a joint work with S.Sivananthan. |
| http://univie.ac.at/projektservice-mathematik/e/talks/Patel_2022-02_Dhiraj Strobl22 Conference.pdf |
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