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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Oblique projections in atomic spaces


Author: Akram Aldroubi
Journal: Proc. Amer. Math. Soc. 124 (1996), 2051-2060
MSC (1991): Primary 41A15, 42C15, 46C99, 47B37
MathSciNet review: 1317028
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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.


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Additional Information

Akram Aldroubi
Affiliation: NIH/BEIP, Building 13/3N17, 13 South DR MSC 5766, Bethesda, Maryland 20892-5766
Email: aldroubi@helix.nih.gov

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03255-8
PII: S 0002-9939(96)03255-8
Keywords: Oblique projection, biorthogonal multiwavelet, multiwavelets, unitary operators, Riesz basis
Received by editor(s): January 3, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society