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 {M_n} denote the n n -fold convolution of a compactly supported function M ∈ L 2 ( R d ) M \in {L_2}({{\mathbf {R}}^d}) and denote by \[ S n := { ∑ j ∈ Z d c ( j ) M n ( ⋅ − j ) : c ∈ l 2 ( Z d ) } {S_n}: = \left \{ {\sum \limits _{j \in {{\mathbf {Z}}^d}} {c(j){M_n}( \cdot - j):c \in {l_2}({{\mathbf {Z}}^d})} } \right \} \] the span of the translates of M n {M_n} . We prove that there exists a set Ω \Omega with <inline-formula content-type="mat