Explicit localization estimates for spline-type spaces

holds for every c ∈ `, and the functions {fk}k satisfy an spatial localization condition. In a spline-type space any function in f ∈ S has a unique expansion f = ∑ k ckfk. Moreover the coefficients are given by ck = 〈f, gk〉, where {gk}k ⊆ S is the dual basis, a set of functions characterized by the relation 〈gk, fj〉 = δk,j . These spaces provide a very natural framework for the sampling problem. The general theory of localized frames (see [6], [5] and [2]) asserts that the functions forming the dual basis satisfy a similar spatial localization. This can be used to extend the expansion in (1) to other spaces, so that the family {fk}k becomes a Banach frame for an associated family of Banach spaces (see [4] and [6]). In the case of a splinetype space S, this means that the decay of a function in S can be characterized by the decay of its coefficients and, in particular, that the functions {fk}k form a so called pRiesz basis for its L-closed linear span, for the whole range 1 ≤ p ≤ ∞. We derive, in some concrete case, explicit bounds for the localization of the dual basis. We will work with a set of functions satisfying a polynomial decay condition around a set of nodes forming a lattice. By a change of variables, we can assume that the lattice is Z. So, we will consider a set of functions {fk}k ⊆ L(R) satisfying the condition, |fk(x)| ≤ C (1 + |x− k|) , x ∈ R and k ∈ Z,