Oblique projections in atomic spaces

Let ${\mathcal H}$ be a Hilbert space, $\mathbf O$ a unitary operator on ${\mathcal H}$, and
$\{\phi ^i\}_{i=1,\dots ,r.}$ $r$ vectors in ${\mathcal H}$. We construct an atomic subspace $U \subset {\mathcal H}$:

We give the necessary and sufficient conditions for $U$ to be a well-defined, closed subspace of ${\mathcal H}$ with $\left \{ {\mathbf O ^k\phi ^i} \right \}_{i=1,\dots ,r, \;k\in \mathbf Z }$ as its Riesz basis. We then consider the oblique projection $\mathbf P _{{\scriptscriptstyle U\bot V}}$ on the space $U(\mathbf O ,\{\phi ^i_{\scriptscriptstyle U}\}_{i=1,\dots ,r})$ in a direction orthogonal to $V(\mathbf O ,\{\phi ^i_{\scriptscriptstyle V}\}_{i=1,\dots ,r})$. We give the necessary and sufficient conditions on $\mathbf O ,\{\phi ^i_{\scriptscriptstyle U}\}_{i=1,\dots ,r}$, and $\{\phi ^i_{\scriptscriptstyle V}\}_{i=1,\dots ,r}$ for $\mathbf P _{{\scriptscriptstyle U\bot V}}$ 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.