Characterizations of multivariate differences and associated exponential splines

The subject of this investigation is the class of difference functionals—linear combinations of finitely many function and/or derivative evaluations—which annihilate the nullspace of a certain constant coefficient differential operator. Any such functional can be viewed as an integral-differential operator whose Peano kernel is a compactly supported exponential spline. Besides extending some earlier results [15], we show that these functionals are the only ones whose convolutions with the associated exponential truncated powers have compact support. It is then proven that, in case the functional depends entirely on function values at rational points, it must be a linear combination of forward differences. These results have applications in the areas of (a) placing compactly supported exponential splines in the span of the box splines, and (b) interpolation by exponential polynomials to function values at the support points of the forward difference functional.