Singularity of Cardinal Interpolation with Shifted Box Splines Original Research Article

Cardinal interpolation by integer translates of shifted box splines Mn, ω ≔ Mnnn(· + ω) on the three-direction mesh is studied. Let Λ ≔ (−12, 12)2 ∩ {(s, t) : |s − t| < 12}. In a previous work by these authors it was shown that the symbol of Mn, ω does not vanish on the torus T2 for all ω in the shift region Λ. In this work, it is shown that the symbol of Mn, ω always vanishes somewhere on T2 if ω ∈ [−12, 12]2\Λ. In other words the cardinal interpolation operator corresponding to Mn, ω, ω ∈ [−12, 12]2, n = 1, 2, ..., is invertible if and only if ω ∈ Λ.