Coefficient Quantization in Banach Spaces |
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| Authors: | S J Dilworth E Odell T Schlumprecht A Zsák |
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| Institution: | (1) Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA;(2) Department of Mathematics, The University of Texas, 1 University Station C1200, Austin, TX 78712, USA;(3) Department of Mathematics, Texas A&M University, College Station, TX 78712, USA;(4) School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK |
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| Abstract: | Let (e
i
) be a dictionary for a separable infinite-dimensional Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn
usually from a ‘finite alphabet’. We investigate several approximation properties of this type and connect them to the Banach
space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c
0. We also show that, for every ε>0, the unit ball of every separable infinite-dimensional Banach space X contains a dictionary (x
i
) such that the additive group generated by (x
i
) is (3+ε)−1-separated and 1/3-dense in X.
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| Keywords: | Coefficient quantization Banach spaces Biorthogonal systems Σ – Δ algorithm Containment of c 0 Lattices |
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