Universal series induced by approximate identities and some relevant applications

We prove the existence of series ∑a_nψ_n, whose coefficients (a_n) are in ∩_p>1?^p and whose terms (ψ_n) are translates by rational vectors in R^d 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 W(C₀,?¹), C_b(R^d), C₀(R^d), L^p(R^d), for every p∈1,∞), 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.