Convergence of cardinal series

Abstract:

The result of this paper is a generalization of our characterization of the limits of multivariate cardinal splines. Let ${M_n}$ denote the $n$-fold convolution of a compactly supported function $M \in {L_2}({{\mathbf {R}}^d})$ and denote by \[ {S_n}: = \left \{ {\sum \limits _{j \in {{\mathbf {Z}}^d}} {c(j){M_n}( \cdot - j):c \in {l_2}({{\mathbf {Z}}^d})} } \right \}\] the span of the translates of ${M_n}$. We prove that there exists a set $\Omega$ with ${\operatorname {vol} _d}(\Omega ) = {(2\pi )^d}$ such that for any $f \in {L_2}({{\mathbf {R}}^d})$, \[ \operatorname {dist} (f,{S_n}) \to 0\quad {\text {as }}n \to \infty ,\] if and only if the support of the Fourier transform of $f$ is contained in $\Omega$.