Given a first-kind integral equation \[ K u ( x ) = ∫ 0 1 K ( x , t ) u ( t ) d t = f ( x ) \mathcal {K}u(x) = \int _0^1 {K(x,t)u(t)\,dt = f(x)} \] with discrete noisy data d i = f ( x i ) + ε i {d_i} = f({x_i}) + {\varepsilon _i} , i = 1 , 2 , … , n i = 1,2, \ldots ,n , let u n α {u_{n\alpha }} be the minimizer in a Hilbert space W of the regularization functional ( 1 / n ) ∑ ( K u ( x i ) − d i ) 2 + α ‖ u ‖ W 2 (1/n)\sum {(\mathcal {K}} u({x_i}) - {d_i}{)^2} + \alpha \left \| u \right \|_W^2 . It is shown that in any one of a wide class of norms, which includes ‖ ⋅ ‖ W {\left \| \cdot \right \|_W} , if α → 0 \alpha \to 0 in a certain way as n → ∞ n \to \infty , then u n α {u_{n\alpha }} converges to the true solution u 0 {u_0} . Convergence rates are obtained and are used to estimate the optimal regularization parameter α \alpha .