p-Frames in Separable Banach Spaces |
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Authors: | Ole Christensen Diana T Stoeva |
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Institution: | (1) Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark;(2) Department of Mathematics, University of Chemical Technology and Metallurgy, Blvd. Kl. Ohridski 8, 1756 Sofia, Bulgaria |
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Abstract: | Let X be a separable Banach space with dual X
*. A countable family of elements {g
i
}X
* is a p-frame (1 p ) if the norm
X
is equivalent to the
p
-norm of the sequence {g
i
()}. Without further assumptions, we prove that a p-frame allows every gX
* to be represented as an unconditionally convergent series g=d
i
g
i
for coefficients {d
i
}
q
, where 1/p+1/q=1. A p-frame {g
i
} is not necessarily linear independent, so {g
i
} is some kind of overcomplete basis for X
*. We prove that a q-Riesz basis for X
* is a p-frame for X and that the associated coefficient functionals {f
i
} constitutes a p-Riesz basis allowing us to expand every fX (respectively gX
*) as f=g
i
(f)f
i
(respectively g=g(f
i
)g
i
). In the general case of a p-frame such expansions are only possible under extra assumptions. |
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Keywords: | p-frame p-Riesz basis Banach space |
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