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Projections and dyadic Parseval frame MRA wavelets. (English) Zbl 1330.42023
This paper studies dyadic MRA wavelets in \(L^2(\mathbb R)\). It is a continuation and refinement of the articles by 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. D. Han and D. R. Larson [Mem. Am. Math. Soc. 697, 94 p. (2000; Zbl 0971.42023)], W. Czaja [Proc. Am. Math. Soc. 136, No. 3, 867–871 (2008; Zbl 1136.42027)]).

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
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