Ces\`aro summability of H\"older functions and Talbot effect on rank one Riemannian symmetric spaces of compact type

TL;DR

This paper characterizes H"older continuity via Ces extbackslash ar extbackslash o means on rank one symmetric spaces, and applies this to analyze the Talbot effect and Schr"odinger propagation.

Contribution

It provides a new quantitative characterization of H"older continuity on symmetric spaces and connects it to diffraction phenomena like the Talbot effect.

Findings

Characterization of H"older continuity using Ces extbackslash ar extbackslash o means.

Analysis of the Talbot effect through Schr"odinger propagation.

Oscillatory expansions of zonal spherical functions near key points.

Abstract

On rank one Riemannian symmetric spaces of compact type (of dimension $\ge 2$), we first obtain a quantitative characterization of H\"older continuity in terms of Ces\`aro means. In addition to some approximation theoretic applications, we also apply it to study the celebrated physical phenomenon known as `Talbot effect' arising from diffraction theory. More precisely, for almost every fixed time instance, we study the H\"older continuity and the fractal profile of the Schr\"odinger propagation in terms of the decay of the Littlewood-Paley projections of the initial data. In the process, we also obtain oscillatory expansions of zonal spherical functions uniformly near the origin and near the cut locus respectively, which may be of independent interest.