On the convergence of regular families of cardinal interpolators

Abstract

In this note a general way to develop a cardinal interpolant for l 2 -data on the integer lattice Z n is shown. Further, a parameter is introduced which allows one to recover the original Paley-Weiner function from which the data came.

Convergence Properties of Spline-Like Cardinal Interpolation Operators Acting on $$\ell ^p$$ ℓ p Data

“…We need a definition similar to the one found in [8]. For our purposes, we make the following definition.…”

“…Additionally, in the language of Aldroubi and Unser, all of these examples are more or less interpolating generating functions for L 2 ([−π, π]), see Section 4.3 in [2] for more details. The work done in [8] for the space L 2 suggested that an appropriate generalization to L p should exist.…”

“…For instance, condition (A2) requiresφ α (ξ ) ≥ 0. This condition is used in [8] to prove an interpolation result. However, one could start with a family of functions which satisfies an L 2 interpolating condition, then we could drop this requirement.…”

“…In order to use (10), we continue with a ∈ (R \Ñ ) ∩ {a ∈ R : |a + 1/2| ≥ 1/2}. Condition (B1) is straightforward and checked in [8]. We may recycle most of the estimates above to check (B2), except we must use (10).…”

“…Sufficient conditions on a family of functions to possess the limiting property above were studied by the author in [8], in the context of L 2 cardinal interpolation.…”