Cardinal hermite interpolation with box splines

Abstract

The study of cardinal interpolation (CIP) by the span of the lattice translates of a box spline has met with limited success. Only the case of interpolation with the box spline determined by the three directionsd ₁=(1, 0),d ₂=(0, 1), andd ₃=(1, 1) inR ² has been treated in full generality [2]. In the case ofR ^d,d ≥ 3, the directions that define the box spline must satisfy a certain determinant condition [6], [9]. If the directions occur with even multiplicities, then this condition is also sufficient. For Hermite interpolation (CHIP) both even multiplicities and the determinant condition for the directions does not prevent the linear dependence of the basis functions. This leads to singularities in the characteristic multiplier when using the standard Fourier transform method. In the case of derivatives in one direction, these singularities can be removed and a set of fundamental splines can be given. This gives the existence of a solution to CHIP inL ^p(R ^d) for data inl ^p(Z ^d), 1≤p≤2.