Polynomial approximation in the mean on segments

Let S k S_k , 1 ≤ k ≤ m 1≤ k≤ m , be pairwise disjoint segments, S k = [ a k , b k ] S_k = [a_k, b_k] , 1 > p k > ∞ 1>p_k>∞ . Suppose that functions f k f_k are defined on S k S_k , f k f_k belongs to C ( S k ) C(S_k) and f k ′ f’_k belongs to L P k ( S k ) LP_k(S_k) . It is shown that for n = 1 , 2 , … n=1,2, there are polynomials <mml:math xmlns:mml