Characterizing Wavelet Coefficient Decay of Discrete-Time Signals

We present an intrinsically discrete-time characterization of wavelet coefficient decay. To be more precise, let f = (f(n))n∈Z be a sequence and denote by (dj,l)j≥1,l∈Z the coefficients obtained by passing f through a subsampled wavelet filter bank. Then it is common practice to relate the decay properties of (dj,l) to continuous-time smoothness spaces such as the homogeneous Besov spaces Ḃα p,q(R). We discuss an alternative approach using only discrete-time notions, showing that under suitable assumptions wavelet coefficient decay characterizes precisely the elements of the discrete-time Besov spaces defined by R.H.Torres [12]. The results do not follow as trivial consequences of the continuous-time theory, and seem well adapted to practical applications due to their fully discrete-time nature.