Expansion of Riemann--Liouville process in a critical case

TL;DR

This paper investigates the convergence properties of Haar-based series representations of the critical Riemann--Liouville process, revealing limitations in their optimality in the space of continuous functions.

Contribution

It demonstrates that the Haar-based series representation of the critical Riemann--Liouville process is not rearrangement optimal in convergence rate.

Findings

Haar series representation is not rearrangement optimal for the critical case

Convergence rate limitations are identified in the ${\mathbf C}[0,1]$ space

Highlights the need for alternative representations in critical cases

Abstract

We show that Haar-based series representation of the critical Riemann--Liouville process $R^{\alpha}$ with $\alpha=3/2$ is rearrangement non-optimal in the sense of convergence rate in ${\mathbf C}[0,1]$.