An Analysis of Cardinal Spline-Wavelets

✍ Scribed by C.K. Chui; J.Z. Wang

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 387 KB
• Volume: 72
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1993.1006

✦ Synopsis

The (m) th order cardinal (B)-spline-wavelet (or simply, (B)-wavelet) (\psi_{m}) is known to generate orthogonal decompositions of any function in (L^{2}(-\infty, \infty)). Since (\psi_{m}) is usually .considered as a bandpass filter, a wavelet series (g=\sum c, \psi_{m}(\cdots)) 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 (m) th order (B)-spline, the (B)-wavelet (\psi_{m}) seems to possess a remarkable property, which we call "complete oscillation." (i) 1993 Academic Press, Inc.