Unconditionality of Periodic Orthonormal Spline Systems in $L^p$

TL;DR

This paper proves that periodic orthonormal spline systems of any order form an unconditional basis in L^p spaces for all p between 1 and infinity, given dense point sequences on the torus.

Contribution

It establishes the unconditional basis property of periodic orthonormal spline systems in L^p spaces for all p in (1,∞), extending previous results to a broader setting.

Findings

Periodic orthonormal spline systems are unconditional bases in L^p for 1<p<∞.

The result holds for any natural number order k and dense point sequences on the torus.

The proof applies to a wide class of spline systems in harmonic analysis.

Abstract

Given any natural number $k$ and any dense point sequence $(t_n)$ on the torus $\mathbb T$, we prove that the corresponding periodic orthonormal spline system of order $k$ is an unconditional basis in $L^p$ for $1<p<\infty$.