Convergence of cardinal series

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 ∈ {L_2}({{R}^d}) and denote by \[ S n := { ∑ j ∈ Z d c ( j ) M n ( ⋅ − j ) : c ∈ l 2 ( Z d ) } {S_n}: = \{ {∑ _{j ∈ {{Z}^d}} {c(j){M_n}( · - j):c ∈ {l_2}({{Z}^d})} } \} \] the span of the translates of M n {M_n} . We prove that there exists a set Ω Ω with <inline-formula content-type="mat