Characterizing wavelet coefficient decay of discrete-time signals

Abstract

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 ( d j , 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 ( d j , l ) to continuous-time smoothness spaces such as the homogeneous Besov spaces B ˙ p , q α ( 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 in Spaces of sequences, sampling theorem and functions of exponential type, Studia Math. 100 (1991) 51–74. 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.