An Analysis of Cardinal Spline-Wavelets Original Research Article

The mth order cardinal B-spline-wavelet (or simply, B-wavelet) ψm is known to generate orthogonal decompositions of any function in L2(−∞, ∞). Since ψm is usually considered as a bandpass filter, a wavelet series g = ∑ cjψm(·−j) may be treated as a bandpass signal. Hence, the problem of characterizing g from its "zerocrossings" is very important in the application of spline-wavelets to signal analysis. However, since g is not an entire function, weak sign changes of g must also be taken into consideration. The objective of this paper is to initiate a study of this important problem. It is noted, in particular, that in contrast to the total positivity property of the mth order B-spline, the B-wavelet ψm seems to possess a remarkable property, which we call "complete oscillation."