Coefficient Quantization in Banach Spaces Academic Article uri icon

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. © 2007 SFoCM.

author list (cited authors)

  • Dilworth, S. J., Odell, E., Schlumprecht, T., & Zsák, A.

citation count

  • 5

publication date

  • August 2007