Density and stability of wavelet frames

✍ Scribed by Wenchang Sun; Xingwei Zhou

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 253 KB
• Volume: 15
• Category: Article
• ISSN: 1063-5203
• DOI: 10.1016/s1063-5203(03)00049-6

✦ Synopsis

Density conditions including necessary ones and sufficient ones for irregular wavelet systems to be frames are studied in this paper. We give a definition of affine Beurling density of time-scale parameters and show that for irregular wavelet systems to be Bessel sequences, the time-scale parameters must be relatively uniformly discrete, or equivalently, they must have a finite upper affine Beurling density. We also show that the lower affine Beurling density of time-scale parameters must be positive for a large class of wavelet frames. For sufficient conditions, we prove that for a nice function, every relatively uniformly discrete and (a, b)-dense time-scale sequence will generate a frame if a -1 and b are small enough. Explicit frame bounds are given. We also study the stability of wavelet frames and show that every wavelet frame with arbitrary time-scale parameters is stable provided the generating function is nice enough. Explicit stability bounds are given. Numerical examples show that our results are sharper than some known ones.