Nonhomogeneous wavelet systems in high dimensions

Abstract

Continuing the lines developed in Han (2010) [20], in this paper we study nonhomogeneous wavelet systems in high dimensions. It is of interest to study a wavelet system with a minimum number of generators. It has been shown in Dai et al. (1997) [9] that for any d×d real-valued expansive matrix M, a homogeneous orthonormal M-wavelet basis can be generated by a single wavelet function. On the other hand, it has been demonstrated in Han (2010) [20] that nonhomogeneous wavelet systems, though much less studied in the literature, play a fundamental role in wavelet analysis and naturally link many aspects of wavelet analysis together. In this paper, we are interested in nonhomogeneous wavelet systems in high dimensions with a minimum number of generators. As we shall see in this paper, a nonhomogeneous wavelet system naturally leads to a homogeneous wavelet system with almost all properties preserved. We also show that a nonredundant nonhomogeneous wavelet system is naturally connected to refinable structures and has a fixed number of wavelet generators. Consequently, it is often impossible for a nonhomogeneous orthonormal wavelet basis to have a single wavelet generator. However, for redundant nonhomogeneous wavelet systems, we show that for any d×d real-valued expansive matrix M, we can always construct a nonhomogeneous smooth tight M-wavelet frame in L2(Rd) with a single wavelet generator whose Fourier transform is a compactly supported C∞ function. Moreover, such nonhomogeneous tight wavelet frames are associated with filter banks and can be easily modified to achieve directionality in high dimensions. As an illustration, we present a bivariate directional tight 2I2-wavelet frame, which has an underlying tight framelet filter bank and can be efficiently implemented. Our analysis of nonhomogeneous wavelet systems employs a notion of frequency-based nonhomogeneous (or more generally, nonstationary) dual wavelet frames in the distribution space. Such a notion allows us to completely separate the perfect reconstruction property of a wavelet system from its stability in various function spaces. We provide a complete characterization of frequency-based nonstationary dual wavelet frames in the distribution space.