Convolution and product in finitely generated shift-invariant spaces

We discuss some structural properties of finitely generated shift-invariant (FGSI) spaces and subspaces of Sobolev spaces, particularly those related to convolution and the product within these spaces. We find shift-invariant solutions in FGSI spaces for a class of differential-difference equations with constant coefficients. Additionally, we analyze the Fourier multipliers in FGSI spaces and the wave fronts for the convolution and product in FGSI spaces.