The Fourier Series of a Sequence of Functions

Let f_k(x)εL^r, r≥1, in 0≤2π, k=1, 2... . Let a_n^(k)→A_n, b_n^(k)→B_n, as k→∞ for each n. A necessary and sufficient condition that (A_n, B_n) should be the Fourier series of a function F(x)εL^r which is the limit in the mean of {f_k(x)} is that

lim lim x^(k)_n=0.