p-Frames in Separable Banach Spaces

被引:0
作者
Ole Christensen
Diana T. Stoeva
机构
[1] Technical University of Denmark,Department of Mathematics
[2] University of Chemical Technology and Metallurgy,Department of Mathematics
来源
Advances in Computational Mathematics | 2003年 / 18卷
关键词
-frame; -Riesz basis; Banach space;
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摘要
Let X be a separable Banach space with dual X*. A countable family of elements {gi}⊂X* is a p-frame (1 p ∞) if the norm ‖⋅‖X is equivalent to the ℓp-norm of the sequence {gi(⋅)}. Without further assumptions, we prove that a p-frame allows every g∈X* to be represented as an unconditionally convergent series g=∑digi for coefficients {di}∈ℓq, where 1/p+1/q=1. A p-frame {gi} is not necessarily linear independent, so {gi} 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 {fi} constitutes a p-Riesz basis allowing us to expand every f∈X (respectively g∈X*) as f=∑gi(f)fi (respectively g=∑g(fi)gi). In the general case of a p-frame such expansions are only possible under extra assumptions.
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页码:117 / 126
页数:9
相关论文
共 14 条
[1]  
Aldroubi A.(2001)Non-uniform sampling in shift invariant spaces SIAM Rev. 43 585-620
[2]  
Gröchenig K.H.(2001)p-frames and shift invariant subspaces of J. Fourier Anal. Appl. 7 1-22
[3]  
Aldroubi A.(1999)Frames for Banach spaces Contemp. Math. 247 149-182
[4]  
Sun Q.(2001)Frames, bases, and discrete Gabor/wavelet expansions Bull. Amer. Math. Soc. 38 273-291
[5]  
Tang W.(1991)Describing functions: frames versus atomic decompositions Monatshefte für Mathematik 112 1-41
[6]  
Casazza P.(1989)Continuous and discrete wavelet transforms SIAM Rev. 31 628-666
[7]  
Han D.(1997)New characterizations of Riesz bases Appl. Comput. Harmon. Anal. 4 222-229
[8]  
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