A generalization of the rogosinski-rogosinski theorem - Dekanov, S. Ya.; Mikhalin, G. A. (Ukrains’kyi Matematychnyi Zhurnal, 2000)
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 \(_U_r olimits_j = 1^∞ α _j (x)f_j (x) = a \) yield \(_U_r f_1 (x) = a.\) 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) ≡ αj, and the space L is the m-dimensional Euclidean space.