Riesz Bounds of Spline Affine Systems

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.