Oblique projections in atomic spaces

Let H be a Hilbert space, O a unitary operator on H, and {phi(i)}(i=1,...,tau). tau vectors in H. We construct an atomic subspace U subset of H: [GRAPHICS] We give the necessary and sufficient conditions for U to be a well-defined, closed subspace of H with {O-k phi(i)}(i=1,...,tau, k is an element of Z) consider the oblique projection P-U perpendicular to V on the space U(O,{phi(U)(i)}(i=1,...,tau)) in a direction orthogonal to V(O, {phi(U)i}(i=1,...,tau)). We give the necessary and sufficient conditions on O, {phi(U)(i)}(i=1,...,tau), and {phi(V)(i)}(i=1,...,tau) for P-U perpendicular to 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.