Approximations by convolutions and antiderivatives

Let g be a given function in L ¹ = L ¹(0, 1), and let B be one of the spaces L ^p (0, 1), 1 ≤ p < ∞, or C ₀[0, 1]. We prove that the set of all convolutions f * g, f ∈ B, is dense in B if and only if g is nontrivial in an arbitrary right neighborhood of zero. Under an additional restriction on g, we prove the equivalence in B of the systems f _n * g and I f _n , where f _n ∈ L ¹, n ∈ ?, and I f = f * 1 is the antiderivative of f. As a consequence, we obtain criteria for the completeness and basis property in B of subsystems of antiderivatives of g.