# Riesz Bounds of Spline Affine Systems

## TL;DR

This paper constructs a family of spline affine Riesz bases with bounds independent of the spline order, using Rademacher chaos series and Walsh spectrum analysis to establish their properties.

## Contribution

It introduces a new class of spline affine Riesz bases with uniform bounds, constructed via iterative integration and antiperiodization of spline functions.

## Key findings

- Riesz bounds are independent of the spline parameter m
- Spline functions are represented as finite Rademacher chaos series
- Orthogonality is analyzed through Walsh spectrum

## Abstract

We construct a family of spline affine Riesz bases, i.e. sequences of dilations and translations generated by the special spline functions $\psi_m$, and we prove that their Riesz bounds are independent of $m$. We put $\psi_0=\chi$ as the Haar step function and every following function $\psi_{m+1}$ is obtained by integrating the previous function $\psi_m$ with consequent antiperiodization. We give a representation of the spline $\psi_m$ as a finite sum of Rademacher chaos series and we use a notion of a simple Walsh spectrum of a function in connection with orthogonality of affine systems.