The asymptotic formula for the error in orthogonal projection (Q2785225)

Let \(\phi: \mathbb{R}^d\to \mathbb{R}\) be a continuous function with compact support and \(S\) the closed subspace of \(L^2(\mathbb{R}^d)\) spanned by the integer translates of \(\phi\). Under suitable hypotheses the authors establish the asymptotic formula for the error associated to the orthogonal projection from \(L^2(\mathbb{R}^d)\) to \(S\). The results are applicable to box splines, wavelets or functions based on Strang-Fix construction. Related results have been obtained by \textit{M. Unser} and \textit{I. Daubechies} [IEEE Trans. Sign. Proc. 45, No. 7, 1697-1711 (1997; Zbl 0879.94005)], \textit{C. de Boor}, \textit{R. A. DeVore} and \textit{A. Ron} [Trans. Am. Math. Soc. 341, No. 2, 787-806 (1994; Zbl 0790.41012)] and the authors in previous papers.