The union of translates of a closed unit n-dimensional cube whose edges are parallel to the coordinate unit vectors e,,...,en and whose centers are te,-, |i| < fc, 1 < j' < n, is called a (k, n)-cross. A system of translates of a (fc, n)-cross is called an integer (a rational) lattice tiling if its union is n-space and the interiors of its elements are disjoint, the translates form a lattice an...

In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert’s Nullstellensatz we obtain a necessary condition for tiling n-space by translates of a cluster of cubes. Further, the polynomial method will enable us to show that if there exists a tiling of n-space by translates of a cluster V of prime size then there is a lattice tiling by V as well. Finally, we...

Interpolation by translates of \radial" basis functions is optimal in the sense that it minimizes the pointwise error functional among all comparable quasi{interpolants on a certain \native" space of functions F . Since these spaces are rather small for cases where is smooth, we study the behavior of interpolants on larger spaces of the form F 0 for less smooth functions 0. It turns out that in...