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 e>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+e)−1-separated and 1/3-dense in X.
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