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
.