Oblique projections in atomic spaces

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Abstract
LetH\mathcal {H}be a Hilbert space,O\mathbf {O}a unitary operator onH\mathcal {H}, and{ϕi}i=1,…,r.\{\phi ^i\}_{i=1,\dots ,r.}rrvectors inH\mathcal {H}. We construct anatomic subspaceU⊂HU \subset \mathcal {H}:U={∑i=1,…,r∑k∈Zci(k)Okϕi:ci∈l2,∀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 forUUto be a well-defined, closed subspace ofH\mathcal {H}with{Okϕi}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 projectionPU⊥V\mathbf {P}_{{\scriptscriptstyle U\bot V}}on the spaceU(O,{ϕUi}i=1,…,r)U(\mathbf {O},\{\phi ^i_{\scriptscriptstyle U}\}_{i=1,\dots ,r})in a direction orthogonal toV(O,{ϕVi}i=1,…,r)V(\mathbf {O},\{\phi ^i_{\scriptscriptstyle V}\}_{i=1,\dots ,r}). We give the necessary and sufficient conditions onO,{ϕUi}i=1,…,r\mathbf {O},\{\phi ^i_{\scriptscriptstyle U}\}_{i=1,\dots ,r}, and{ϕVi}i=1,…,r\{\phi ^i_{\scriptscriptstyle V}\}_{i=1,\dots ,r}forPU⊥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.

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