On the zeros of the generalized Euler–Frobenius Laurent polynomial and reconstruction of cardinal splines of polynomial growth from local average samples

Let h(t)h(t) be a nonnegative measurable function supported in [−12,12] and Md(t)=(χ[−12,12]⋆χ[−12,12]⋆⋯⋆χ[−12,12])(t)(d+1 times)(d+1 times) be the central B-spline of degree d  . We show that the roots of the generalized Euler–Frobenius Laurent polynomial defined by Eh,d(z):=∑n∈Z(h⋆Md)(n)znEh,d(z):=∑n∈Z(h⋆Md)(n)zn are simple, negative and all are different from −1. As a consequence of this result, we show that for every sequence {yn}n∈Z∈RZ{yn}n∈Z∈RZ of samples having polynomial growth and nonnegative measurable function h   supported in [−12,12], there is a unique spline f of degree d   with polynomial growth satisfying (f⋆h)(n)=yn,n∈Z(f⋆h)(n)=yn,n∈Z. The presented work answers affirmatively the open problem posed in Pérez-Villalón and Portal (2012) [9].