8th International Conference on
Computational Harmonic Analysis

September 12-16, 2022

Ingolstadt, Germany

"Marcinkiewicz--Zygmund inequalities for scattered and random data on the $q$-sphere"

Jahn, Thomas

The recovery of functions and estimating their integrals from finitely many samples is one of the central tasks in approximation theory. Marcinkiewicz--Zygmund (MZ) inequalities provide answers to both the recovery and the quadrature aspect. In this talk, we put ourselves on the $q$-dimensional sphere $\mathbb{S}^q$, and investigate the validity of the MZ inequality $$ (1-\eta)\|f\|_{\sigma_q,p}\leq \left(\sum_{j=1}^N w_j |f(\xi_j)|^p \right)^{\frac{1}{p}}\leq (1+\eta)\|f\|_{\sigma_q,p} $$ for polynomials $f$ of maximum degree $n$ on the sphere $\mathbb{S}^q$, subject to weights $w_j$, and the number and distribution of the (deterministic or randomly chosen) sample points $\xi_1,\ldots,\xi_N$ on $\mathbb{S}^q$.
http://univie.ac.at/projektservice-mathematik/e/talks/ICCHA2022_Jahn_2022-04_ICCHA2022 - Jahn.pdf

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