Coefficient Quantization in Banach Spaces

Let (ei) be a dictionary for a separable 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 c0.