Let
H
\mathcal {H}
be a Hilbert space,
O
\mathbf {O}
a unitary operator on
H
\mathcal {H}
, and
{
ϕ
i
}
i
=
1
,
…
,
r
.
\{\phi ^i\}_{i=1,\dots ,r.}
r
r
vectors in
H
\mathcal {H}
. We construct an atomic subspace
U
⊂
H
U \subset \mathcal {H}
:
U
=
{
∑
i
=
1
,
…
,
r
∑
k
∈
Z
c
i
(
k
)
O
k
ϕ
i
:
c
i
∈
l
2
,
∀
i
=
1
,
…
,
r
}
.
\begin{equation*} U=\left \{ { \sum \limits _{i=1,\dots ,r} {\sum \limits _{k\in \mathbf {Z}} {c^i(k)\mathbf {O}^k\phi ^i}:\;c^i\in l^2,\forall i=1,\dots ,r}} \right \}. \end{equation*}
We give the necessary and sufficient conditions for
U
U
to be a well-defined, closed subspace of
H
\mathcal {H}
with
{
O
k
ϕ
i
}
i
=
1
,
…
,
r
,
k
∈
Z
\left \{ {\mathbf {O}^k\phi ^i} \right \}_{i=1,\dots ,r, \;k\in \mathbf {Z}}
as its Riesz basis. We then consider the oblique projection
P
U
⊥
V
\mathbf {P}_{{\scriptscriptstyle U\bot V}}
on the space
U
(
O
,
{
ϕ
U
i
}
i
=
1
,
…
,
r
)
U(\mathbf {O},\{\phi ^i_{\scriptscriptstyle U}\}_{i=1,\dots ,r})
in a direction orthogonal to
V
(
O
,
{
ϕ
V
i
}
i
=
1
,
…
,
r
)
V(\mathbf {O},\{\phi ^i_{\scriptscriptstyle V}\}_{i=1,\dots ,r})
. We give the necessary and sufficient conditions on
O
,
{
ϕ
U
i
}
i
=
1
,
…
,
r
\mathbf {O},\{\phi ^i_{\scriptscriptstyle U}\}_{i=1,\dots ,r}
, and
{
ϕ
V
i
}
i
=
1
,
…
,
r
\{\phi ^i_{\scriptscriptstyle V}\}_{i=1,\dots ,r}
for
P
U
⊥
V
\mathbf {P}_{{\scriptscriptstyle U\bot V}}
to be well defined. The results can be used to construct biorthogonal multiwavelets in various spaces. They can also be used to generalize the Shannon-Whittaker theory on uniform sampling.