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Approximate identities for the Schwartz space

Abstract:

We establish a sufficient and a necessary conditions for the convergence, in the Schwartz space topology, of the sequence obtained by the convolution of an arbitrary given approximate identity for L(Rk) , whose terms belongs to the Schwartz space S(Rk) , with any arbitrary Schwartz function to this function. Additionally, we give an example of a sequence, (ψn)n=1+∞ in S(Rk) , that is an approximate identity for L(Rk) but for which there exists a function f∈ S(Rk) such that ψn∗ f does not converge to f as n→ + ∞ with respect to the topology of S(Rk).