Oblique projections in atomic spaces

Abstract:

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 $U \subset \mathcal {H}$: \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 $U$ 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.