Reconstruction of splines from local average samples

We study the reconstruction of cardinal splines f(t), from their average samples yn = f ∗h(n), n ∈ Z, when the average function h(t) has support in [−1/2, 1/2]. We investigate the existence and uniqueness of the solution of the following problem: for given dates yn, find a cardinal spline f(t), of a given degree, satisfying yn = f ∗ h(n), n ∈ Z. 1 Spline interpolation and average sampling First, let us introduce some notation and recall some of the fundamental results of Schoenberg’s magnificent cardinal interpolation spline theory (see [4]). Let βd be the cardinal central B-spline of degree d, βd := X[−1/2,1/2] ∗ . . . ∗ X[−1/2,1/2] (d+ 1 terms) and let Sd the space generated by the shift of βd, i.e. the set of functions f(t) that admit a representation of the form f(t) = ∑ n∈Z an βd(t− n) with appropriate coefficients an. When d is odd, the space Sd is the set of cardinal splines of degree d with knot sequence Z, i.e. the set of functions f ∈ Cd−1(R) such that for all k ∈ Z, f |[k,k+1) belongs to Πd the class of polynomials of degree not exceeding d. When d is even, the knot sequence is Z + 1/2, i.e. Sd := { f ∈ Cd−1(R) : f |[k−1/2,k+1/2) ∈ Πd, k ∈ Z } . We consider the cardinal spline interpolation problem: Given a sequence of real numbers {yn}n∈Z, find a spline f ∈ Sd such that f(n) = yn, n ∈ Z. For d = 1 the problem has a unique solution, the linear spline obtained by linear interpolation between every pair of consecutive data, but for d > 1 it has infinitely many solutions forming a linear manifold in Sd of dimension d−1 when d is odd, and of dimension d when d is even. Growth conditions give the uniqueness. Specifically, denoting Sd,γ := { f(t) ∈ Sd : f(t) = (|t|) as t 7→ ±∞ } , Dγ := { {yn}n∈Z : yn = O(|n|) as n 7→ ±∞ } , for γ ≥ 0, Shoenberg proved that for a given sequence of real numbers {yn}n∈Z ∈ Dγ , the problem: Find a spline f ∈ Sd,γ satisfying f(n) = yn, n ∈ Z, ∗E-mail: [email protected] †E-mail: [email protected]