Nonhomogeneous Wavelet Systems in High Dimensions

Abstract

It is of interest to study a wavelet system with a minimum number of generators. It has been showed by X. Dai, D. R. Larson, and D. M. Speegle in [11] that for any $d\times d$ 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 [21] 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 $d\times d$ real-valued expansive matrix M, we can always construct a nonhomogeneous smooth tight M-wavelet frame in $L_2(R^d)$ with a single wavelet generator whose Fourier transform is a compactly supported $C^\infty$ function. Moreover, such nonhomogeneous tight wavelet frames are associated with filter banks and can be modified to achieve directionality in high dimensions. Our analysis of nonhomogeneous wavelet systems employs a notion of frequency-based nonhomogeneous wavelet systems in the distribution space. Such a notion allows us to separate the perfect reconstruction property of a wavelet system from its stability in function spaces.