Universal series induced by approximate identities and some relevant applications

We prove the existence of series [Formula: see text], whose coefficients [Formula: see text] are in [Formula: see text] and whose terms [Formula: see text] are translates by rational vectors in [Formula: see text] of a family of approximations to the identity, having the property that the partial sums are dense in various spaces of functions such as Wiener's algebra [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], for every [Formula: see text], and the space of measurable functions. Applying this theory to particular situations, we establish approximations by such series to solutions of the heat and Laplace equations as well as to probability density functions.