A multivariate analog of Marsden's identity and a quasi-interpolation scheme

Abstract

Let ϕ be a linear combination of certain box splines and$\hat \phi$ its Fourier transform, such that$\hat \phi \left( 0 \right) \ne 0$ and$D^\beta \hat \phi \left( {2\pi k} \right) = 0$ for all κ∈Z^N{0} and β≤α. In this paper we construct an expression of the multivariate polynomial (·-y)^α in terms of a linear combination of the integer translates of ϕ(·), where the coefficients can be computed recursively using only the information on$D^\beta \hat \phi \left( 0 \right)$, β ≤ α. As an application, a quasi-interpolation scheme based only on function values on (scaled) integers κ∈Z^N is constructed that gives a “multivariate order” of approximation that includes both coordinate and total orders.