A generalization of the rogosinski-rogosinski theorem

Abstract

We establish necessary and sufficient conditions for numerical functions αj(x), j ∈ N, x ∈ X, under which the conditions K(f _j ⊂ K(f ₁) ∀j≥2 and \(\mathop {\lim }\limits_{U_r } \sum\nolimits_{j = 1}^\infty {\alpha _j (x)f_j (x) = a} \) yield \(\mathop {\lim }\limits_{U_r } f_1 (x) = a.\) The functions fj(x) are uniformly bounded on the set X and take values in a boundedly compact space L, and K(fj) is the kernel of the function fj. The well-known Rogosinski-Rogosinski theorem follows from the proved statements in the case where X = N, α _j (x) ≡ α_j, and the space L is the m-dimensional Euclidean space.