An optimal result for sampling density in shift-invariant spaces generated by Meyer scaling function

For a class of continuously differentiable function ϕ satisfying certain decay conditions, it is shown that if the maximum gap δ:=supi⁡(xi+1−xi) between the consecutive sample points is smaller than a certain number B0, then any f∈V(ϕ) can be reconstructed uniquely and stably. As a consequence of this result, it is shown that if δ<1, then {xi:i∈Z} is a stable set of sampling for V(ϕ) with respect to the weight {wi:i∈Z}, where wi=(xi+1−xi−1)/2 and ϕ is the scaling function associated with Meyer wavelet. Further, the maximum gap condition δ<1 is sharp.