Let
X
X
and
Y
Y
denote real Hilbert spaces, and let
L
:
X
→
Y
L:\,X \to Y
be a closed densely-defined linear operator having closed range. Given an element
y
∈
Y
y \in Y
, we determine least squares solutions of the linear equation
L
x
=
y
Lx = y
by using the method of regularization. Let
Z
Z
be a third Hilbert space, and let
T
:
X
→
Z
T:\,X \to Z
be a linear operator with
D
(
L
)
⊆
D
(
T
)
\mathcal {D}(L) \subseteq \mathcal {D}(T)
. Under suitable conditions on
L
L
and
T
T
and for each
α
≠
0
\alpha \ne 0
, we show that there exists a unique element
x
α
∈
D
(
L
)
{x_\alpha } \in \mathcal {D}(L)
which minimizes the functional
G
α
(
x
)
=
‖
L
x
−
y
‖
2
+
α
2
‖
T
x
‖
2
{G_\alpha }(x) = {\left \| {Lx - y} \right \|^2} + {\alpha ^2}{\left \| {Tx} \right \|^2}
, and the
x
α
{x_\alpha }
converge to a least squares solution
x
0
{x_0}
of
L
x
=
y
Lx = y
as
α
→
0
\alpha \to 0
. We apply our results to the special case where
L
L
is an
n
n
th-order differential operator in
X
=
L
2
[
a
,
b
]
X = {L^2}[a,b]
, and we regularize using for
T
T
an
m
m
th-order differential operator in
L
2
[
a
,
b
]
{L^2}[a,b]
with
m
≤
n
m \le n
. Using an approximating space of Hermite splines, we construct numerical solutions to
L
x
=
y
Lx = y
by the method of continuous least squares and the method of discrete least squares.