This paper is concerned with the study of the error estimate of n-approximation number an (Tk: Lq ~ Lq), where Tk : Lq[0, 1] - Lq[0, 1] (q >=1), (TK)(s) = 1 0 K(s,t)x(t)dt, s E [0, 1], and kernal K(s, t) E W p r ([0, 1]2). When the result of study is applied to optimal error estimate (Bakhvalov [5]) of solving the second linear Fredholm equation (I - TK)x - y using degenerate method, it produces an optimal error estimate which optimizes that in [10] when q = 1.