Construction of Compactly SupportedM-Band Wavelets

Abstract

In this paper, we consider the asymptotic regularity of Daubechies scaling functions and construct examples of M-band scaling functions which are both orthonormal and cardinal for M ¢ 3. ᭧ 1999 Academic Press

M-Band Scaling Functions with Minimal Support Are Asymmetric

“…For M ≥ 3 and N ≥ 1, similar scaling function with minimal support, which we denoted by φ M,N , was constructed in [8]. It was shown that φ M,N is just the Haar function for N = 1 (hence symmetric for N = 1), has asymptotic Hölder regularity proportionally to ln(sin Mπ/(2M + 2)) −1 (ln M) −1 N for even M and to (4 ln M) −1 ln N for odd M as N tends to infinity [2,11,13], and has explicit expression if N ≤ M − 1 [3,14]. In this note, we show that the M-band scaling function φ M,N with minimal support is asymmetric for N ≥ 2.…”

M-Band Burt–Adelson Biorthogonal Wavelets

“…Our construction can be compared with the one carried out by Chui and Lian in [6], which led to 3-band orthonormal symmetric-antisymmetric wavelets with possibly arbitrarily high regularity and also to two recent constructions of families of M-band wavelets by Belogay and Wang [2] and by Bi et al [3]. However, all these constructions were carried out in the orthogonal case and suffer some of the drawbacks of that setting.…”

“…As explained before, we can consider the restrictions 3 From this one easily deduces that the spectral radius of 3 T |P a | 2 | F 2 , and hence of 3 T |P a | 2 , is…”

“…Chui and Lian present in [6] some families of 3-band symmetric-antisymmetric orthonormal wavelets with good (and possibly arbitrarily high) regularity. Bi et al and Sun presented in [3], an M-band orthonormal, compactly supported, cardinal scaling function, for M ≥ 3 (no such scaling functions exist for M = 2). In [2], Belogay and Wang study families of symmetric orthonormal scaling functions for any dilation M: some filters are given in explicit form and the regularity of the associated wavelets is determined.…”

“…The interest in MRA structures with dilation factor greater than 2 [4,14,15,23,30] is motivated by the theory of M-band channel subband coding schemes [3,14,15,27] and by the attempt to obtain sharper time-frequency localization and greater flexibility in the construction of wavelets. The design of the filters is quite different from the classical case (M = 2): it is, in general, more difficult, and the wavelets are no longer determined, in an essentially unique manner, by a pair of biorthogonal MRAs.…”