In this paper, function spaces V ∩ l^Ap(ω) are considered in the context of their multiplicative structure. The space V is determined by conditions on the values of a function in a disk (for example, C^A, Lip_Aα). We denote by l_A ^p(ω) the space of power series such that their Taylor coefficients are p-summable with weight s. For an analytic function Φ acting in a space of this type, we prove the following alternative: either Φ′(z) = 0, or the space is a Banach algebra with respect to pointwise multiplication. For a wide class of weights w, we establish the continuity of the identity embedding mult(V ∩ l^Ap (ω)) → mult l_A ^p. An estimate for the l^p-multiplicative norm of random polynomials is found. This estimate can be considered as an extension of the known result by Salem-Zygmund. Bibliography: 10 titles.