Bernstein-Nikolskii and Plancherel-Polya inequalities in $L_{p}$-norms on non-compact symmetric spaces

TL;DR

This paper extends Bernstein-Nikolskii and Plancherel-Polya inequalities to $L_{p}$-spaces on non-compact symmetric spaces, defining new function spaces and establishing norm estimates and reconstruction properties.

Contribution

It introduces analogs of entire functions of exponential type in $L_{p}$-spaces on non-compact symmetric spaces and proves related inequalities and stability of function reconstruction.

Findings

Defined $L_{p}$-type spaces on symmetric spaces

Established Nikolskii-type inequalities for these functions

Proved stable reconstruction from measure-based measurements

Abstract

By using Bernstein-type inequality we define analogs of spaces of entire functions of exponential type in $L_{p}(X), 1\leq p\leq \infty$, where $X$ is a symmetric space of non-compact. We give estimates of $L_{p}$-norms, $1\leq p\leq \infty$, of such functions (the Nikolskii-type inequalities) and also prove the $L_{p}$- Plancherel-Polya inequalities which imply that our functions of exponential type are uniquely determined by their inner products with certain countable sets of measures with compact supports and can be reconstructed from such sets of "measurements" in a stable way.