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Oblique Projections in Atomic Spaces

Akram Aldroubi
Proceedings of the American Mathematical Society
Vol. 124, No. 7 (Jul., 1996), pp. 2051-2060
Stable URL: http://www.jstor.org/stable/2161491
Page Count: 10
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Oblique Projections in Atomic Spaces
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Abstract

Let H be a Hilbert space, O a unitary operator on H, and {φi}i = 1, ..., r. r vectors in H. We construct an atomic subspace $U \subset \mathcal{H}$: $U = \big\{\sum_{i = 1, \ldots, r} \sum_{k \in \mathbf{Z}} c^i(k)\mathbf{O}^k\phi^i: c^i \in l^2, \forall i = 1, \ldots, r\big\}.$ We give the necessary and sufficient conditions for U to be a well-defined, closed subspace of H with {Okφi}i = 1, ..., r, k ∈ Z as its Riesz basis. We then consider the oblique projection PU ⊥ V on the space U(O,{φi U}i = 1, ..., r) in a direction orthogonal to V(O, {φi V}i = 1, ..., r). We give the necessary and sufficient conditions on O, {φi U}i = 1,..., r, and {φi V}i = 1,..., r for PU ⊥ 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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