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Nonhomogeneous wavelet systems in high dimensions. (English) Zbl 1241.42028
The author explores the connections between nonhomogeneous and homogeneous wavelet systems in ${L}^{2}\left({ℝ}^{d}\right)$. In particular, it is shown that any inhomogeneous wavelet system yields a homogeneous wavelet system together with a sequence of nonhomogeneous wavelet systems with almost all the properties preserved, and that nonredundant nonhomogeneous wavelet systems have a natural connection to refinable structures. Moreover, a pair of frequency-based nonhomogeneous and nonstationary dual wavelet frames are introduced and characterized in ${𝒟}^{\text{'}}\left({ℝ}^{d}\right)$, and it is shown that it is possible to completely separate the perfect reconstruction property of a wavelet system from its stability in various function spaces. As an application of these results, a complete characterization of a pair of nonhomogeneous and nonstationary dual wavelet frames in ${L}^{2}\left({ℝ}^{d}\right)$, as well as a characterization of nonstationary tight wavelet frames in ${L}^{2}\left({ℝ}^{d}\right)$ is obtained. Finally, the author shows that the type of nonhomogeneous tight wavelet frame considered in this article can be associated with filter banks and be easily modified to achieve directionality in all dimensions.
##### MSC:
 42C40 Wavelets and other special systems 42C15 General harmonic expansions, frames 65T60 Wavelets (numerical methods)