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Semi-orthogonal parseval frame wavelets and generalized multiresolution analyses. (English) Zbl 1106.42026
Summary: We study Parseval frame wavelets in \(L^2(\mathbb{R}^d)\) with matrix dilations of the form \((Df)(x)=\sqrt 2f(Ax)\), where \(A\) is an arbitrary expanding \(n\times n\)-matrix with integer coefficients, such that \(|\det A|=2\). In our study we use generalized multiresolution analyses (GMRA) \((V_j)\) in \(L^2(\mathbb{R}^d)\) with dilations \(D\). We describe, in terms of the underlying multiresolution structure, all GMRA Parseval frame wavelets and, a posteriori, all semi-orthogonal Parseval frame wavelets in \(L^2(\mathbb{R}^d)\). As an application, we include an explicit construction of an orthonormal wavelet on the real line whose dimension function is essentially unbounded.

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
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[1] Baggett, L.; Medina, H.; Merrill, K., Generalized multiresolution analyses, and a construction procedure for all wavelet sets in \(\mathbb{R}^n\), J. Fourier anal. appl., 5, 6, 563-573, (1999) · Zbl 0972.42021
[2] D. Bakić, I. Krishtal, E.N. Wilson, Parseval frame wavelets with \(E_n^{(2)}\)-dilations, Appl. Comput. Harmon. Anal. (special issue), in press · Zbl 1090.42020
[3] B. Behera, S. Madan, On a class of band-limited wavelets not associated with an MRA, Preprint · Zbl 1096.42024
[4] Benedetto, J.J.; Li, S., The theory of multiresolution analysis frames and applications to filter banks, Appl. comput. harmon. anal., 5, 389-427, (1998) · Zbl 0915.42029
[5] Bownik, M., The structure of shift-invariant subspaces of \(L^2(\mathbb{R}^n)\), J. funct. anal., 112, 282-309, (2000) · Zbl 0986.46018
[6] M. Bownik, Z. Rzeszotnik, On the existence of multiresolution analysis for framelets, Preprint, 2004 · Zbl 1081.42026
[7] Bownik, M.; Rzeszotnik, Z.; Speegle, D., A characterization of dimension function of orthonormal wavelets, Appl. comput. harmon. anal., 10, 1, 71-92, (2001) · Zbl 0979.42018
[8] Hernández, E.; Weiss, G., A first course on wavelets, (1996), CRC Press Boca Raton, FL · Zbl 0885.42018
[9] Kim, H.O.; Lim, J.O., On frame wavelets associated with multiresolution analyses, Appl. comput. harmon. anal., 10, 61-70, (2001) · Zbl 1022.94001
[10] Paluszyński, M.; Šikić, H.; Weiss, G.; Xiao, S., Tight frame wavelets, their dimension functions, MRA tight frame wavelets and connectivity properties, Adv. comput. math., 18, 297-327, (2003) · Zbl 1018.42020
[11] Papadakis, M., On the dimension function of orthonormal wavelets, Proc. amer. math. soc., 128, 7, 2043-2049, (2000) · Zbl 0956.42022
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