Translates of exponential box splines and their related spaces

Abstract:

Exponential box splines ($EB$-splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space $\mathcal {H}$ spanned by exponential polynomials. This space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients. The main part of this paper is devoted to algebraic analysis of the space ${\mathbf {H}}$ of all entire functions spanned by the integer translates of an $EB$-spline. This investigation relies on a detailed description of $\mathcal {H}$ and its discrete analog $\mathcal {S}$. The approach taken here is based on the observation that the structure of $\mathcal {H}$ is relatively simple when $\mathcal {H}$ is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process. Also, we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of Fourier analysis. Thus, while generalizing the known theory of polynomial box splines, the results here offer a simpler approach and a new insight towards this important special case. We also identify and study in detail several types of singularities which occur only for complex $EB$-splines. The first is when the Fourier transform of the $EB$-spline vanishes at some critical points, the second is when $\mathcal {H}$ cannot be embedded in $\mathcal {S}$ and the third is when ${\mathbf {H}}$ is a proper subspace of $\mathcal {H}$. We show, among others, that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh.