Perturbation Techniques in Irregular Spline - Type Spaces Hans

In this article we study various perturbation techniques in the context of irregular spline-type spaces. We first present the sampling problem in this general setting and prove a general result on the possibility of perturbing sampling sets. This result can be regarded as an spline-type space analogue in the spirit of Kadec’s Theorem for bandlimited functions (see [14] and [15]). We further derive some quantitative estimates on the amount by which a sampling set can be perturbed, and finally prove a result on the existence of optimal perturbations (with the stability of reconstruction being the optimality criterion). Finally, the techniques developed in the earlier parts of the article are used to study the problem of disturbing a basis for a spline-type space, in order to derive a sufficient criterion for a space generated by irregular translations to be a spline-type space. 1. Preliminaries and notation 1.1. Notation. We will work with functions on R, and denote the Euclidean norm of x ∈ R by |x|2, and maximum norm by |x|∞ = max 1≤k≤d |xk|. Definition 1.1. For a complex valued function f : R → C and M ⊆ R, we denote by supM |f | the supremum of the absolute values of f on M , i.e. sup M |f | := sup {|f(x)| : x ∈M} . For δ > 0 and x ∈ R, we define the δ-oscillation of f at x by oscδ(f)(x) := sup {|f(x)− f(y)| : |y − x|2 ≤ δ} . 1.2. p-Riesz bases. Definition 1.2. Let 1 ≤ p < ∞, B a Banach space and Λ a countable index set. We say that a family {fk}k∈Λ in B is a p-Riesz basis for B if the map `(Λ) −→ B (1) (ck)k 7→ ∑