Coefficient Quantization in Banach Spaces
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 epsilon > 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+epsilon)-1-separated and 1/3-dense in X.
coefficient quantization banach spaces biorthogonal systems