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 vol d ( Ω ) = ( 2 π ) d {\operatorname {vol} _d}(\Omega ) = {(2\pi )^d} such that for any f ∈ L 2 ( R d ) f \in {L_2}({{\mathbf {R}}^d}) , \[ dist ⁡ ( f , S n ) → 0 as n → ∞ , \operatorname {dist} (f,{S_n}) \to 0\quad {\text {as }}n \to \infty , \] if and only if the support of the Fourier transform of f f is contained in Ω \Omega .