\(C^{(\infty)}_a\)-almost periodic functions (Q1976030)

There is considered the space \(C^{(\infty)}_a(\mathbb{R})\) with convergence defined by means of the functional \[ D^{(\infty)}_a(f)= \sup_{t\in\mathbb{R}} \sum^\infty_{i=0} a_i|f^{(i)}(t)| \] with \(a_0= 1\), \(a_i> 0\) for all \(i\). The notion of \(C^{(\infty)}_a\)-almost periodicity in \(C^{(\infty)}(\mathbb{R})\) by means of the functional \(D^{(\infty)}_a\) is defined and the properties of \(C^{(\infty)}_a\)-a.p. functions are investigated. There is proved a theorem on approximation of \(C^{(\infty)}_a\)-continuous functions by means of their Steklov functions (i.e., integral means).