A general setting for constrained
L
1
{L^1}
-approximation is presented. Let
U
n
{U_n}
be a finite dimensional subspace of
C
[
a
,
b
]
C[a,b]
and
L
L
be a linear operator from
U
n
{U_n}
to
C
r
(
K
)
(
r
=
0
,
1
)
{C^r}(K)\;(r = 0,1)
where
K
K
is a finite union of disjoint, closed, bounded intervals. For
υ
,
u
∈
C
r
(
K
)
\upsilon ,u \in {C^r}(K)
with
υ
>
u
\upsilon > u
, the approximating set is
U
~
n
(
υ
,
u
)
=
{
p
∈
U
n
:
υ
≤
L
p
≤
u
on
K
}
{\tilde U_n}(\upsilon ,u) = \{ p \in {U_n}:\upsilon \leq Lp \leq u\;{\text {on}}\;K\}
and the norm is
‖
f
‖
w
=
∫
a
b
|
f
|
w
d
x
\|f\|_w = \int _a^b {|f|w\,dx}
where
w
w
a positive continuous function on
[
a
,
b
]
[a,b]
. We obtain necessary and sufficient conditions for
U
~
n
(
υ
,
u
)
{\tilde U_n}(\upsilon ,u)
to admit unique best
‖
⋅
‖
w
\|\;\cdot \;\|_w
-approximations to all
f
∈
C
[
a
,
b
]
f \in C[a,b]
for all positive continuous
w
w
and all
υ
,
u
∈
C
r
(
K
)
(
r
=
0
,
1
)
\upsilon ,u \in {C^r}(K)\;(r = 0,1)
satisfying a nonempty interior condition. These results are applied to several
L
1
{L^1}
-approximation problems including polynomial and spline approximation with restricted derivatives, lacunary polynomial approximation with restricted derivatives, and others.