Oblique projections in atomic spaces

Abstract

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.