Projections and dyadic Parseval frame MRA wavelets (Q890469)

This paper studies dyadic MRA wavelets in \(L^2(\mathbb R)\). It is a continuation and refinement of the articles by \textit{M. Paluszyński} et al. [J. Geom. Anal. 11, No. 2, 311--342 (2001; Zbl 0985.42020); Adv. Comput. Math. 18, No. 2--4, 297--327 (2003; Zbl 1018.42020)]. After introducing the notation to be used, Section 1 provides a terse and clear review of some properties of dyadic Parseval frame MRA wavelets and orthonormal MRA wavelets, including some proofs. It reviews the definition of multiresolution analysis (MRA), low-pass and high-pass filters. One of the main goals of this article is to understand the Parseval frame analogues of orthonormal MRA wavelets. The authors of the two previously mentioned papers began a systematic study of this issue and developed a new notion of low-pass filters, which they called generalized low-pass filters, from which to build Parseval frame analogues of MRA wavelets. The authors point out that this definition of pseudoscaling function is unnecessarily restrictive, introduce the notion of ``generalized scaling function'', and slightly change the definition of low-pass filters. One difference between the present work and the previous two is that there the authors begin with a filter and associate to it a scaling function, whereas here they begin with a scaling function and then associate to it a family of associated low-pass filters; after selecting the desired filter, one can then construct a wavelet essentially as before. Section 2 of the work under review is devoted to studying the properties of these generalized low-pass filters and generalized scaling functions and the construction of dyadic MRA Parseval frame wavelets. Let \(m \bullet g\) denote the function which satisfies \(\widehat{m \bullet g}= m \cdot \widehat{g}\), and let \(\mathcal{P}^{\mathrm{MRA}}\) denote the set of dyadic MRA Parseval frame wavelets. A function \(\phi\) is called maximal if the linear span of the set of its integral translates is not a proper subset of any other principal shift-invariant space. In Section 3 the authors show that any pair \((\phi,m)\) associated to a wavelet in \(\mathcal{P}^{\mathrm{MRA}}\) is actually the orthogonal projection (in the \(L^2(R)\) sense) of some pair \((\phi^{\ast}, m^{\ast})\), where \(\phi^{\ast}\) is a maximal scaling function and \(m^{\ast}\) is its associated low-pass filter. They also give a complete characterization of those sets \(S \subset R\) such that, for such a pair \((\phi^{\ast}, m^{\ast})\), \(\chi_s \bullet \phi^{\ast}\) is also associated to a wavelet in \(\mathcal{P}^{\mathrm{MRA}}\). This result is an analogue of Naimark's theorem that if \(\{v_n: n \in Z\}\) is a Parseval frame on a Hilbert space \(\mathcal{H}\) then \(\mathcal{H}\) can be linearly and isometrically embedded into a larger Hilbert space \(\mathcal{K}\), so that \(\{v_n: n \in Z\}\) is an orthogonal projection of an orthonormal basis in \(\mathcal{K}\) (see, e.g. \textit{D. Han} and \textit{D. R. Larson} [Mem. Am. Math. Soc. 697, 94 p. (2000; Zbl 0971.42023)], \textit{W. Czaja} [Proc. Am. Math. Soc. 136, No. 3, 867--871 (2008; Zbl 1136.42027)]).