Approximation of certain classes of smooth periodic functions of several variables by means of interpolation splines defined over a uniform net (Q1916586)

In this short note without proofs the author states a result for multivariate spline interpolation on the \(m\)-dimensional torus. Here the interpolation grid is of tensor product-type whereas the multivariate splines are considered as linear combinations of translates of lower-dimensional tensor products of Bernoulli functions. The solvability of the interpolation problem depends on a shift parameter as known from the univariate case. Furthermore, an error estimate in the maximum norm is given for functions with prescribed multivariate modulus of continuity.