Riesz Bounds of Spline Affine Systems

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.

15 pages, 2 figures