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Parseval frame scaling sets and MSF Parseval frame wavelets. (English) Zbl 1198.42048
Summary: We consider the Parseval frame (PF) scaling sets and the MSF Parseval frame wavelets (PFWs) in \(L^{2}(\mathbb{R}^d)\) with dilations induced by expanding matrices \(A\) with integer coefficients of arbitrary determinant such that \(|\det A|=2\). We firstly characterize the PF scaling sets, and then provide a method of construction of PF scaling sets. We also prove that all PF scaling sets arise in that way. Finally, by studying the relation between the MSF PFWs and the PF scaling sets, we derive that each PF scaling set \(S\) gives rise to a MSF PFW \(\psi \), and furthermore each MSF PFW whose dimension function is essentially bounded by 1 arises from a PF scaling set and the corresponding PF MRA. Using our results, one can easily construct various PF scaling sets and MSF PFWs.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
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[1] El Naschie, M.S., SO(10) grand unification in a fuzzy setting, Chaos, solitons & fractals, 32, 3, 958-961, (2007)
[2] El Naschie, M.S., Elementary prerequisites for E-infinity, Chaos, solitons & fractals, 30, 3, 579-605, (2006)
[3] El Naschie MS. Symmetry group prerequisites for E-infinity in high energy physics. Chaos, Solitons & Fractals, in press. doi:10.1016/j.chaos.2007.05.006.
[4] El Naschie, M.S., Exceptional Lie groups hierarchy and the structure of the micro universe, Int J nonlinear sci numer simulat, 8, 3, 445-450, (2007)
[5] El Naschie, M.S., A review of applications and results of E-infinity theory, Int J nonlinear sci numer simulat, 8, 1, 11-20, (2007)
[6] El Naschie, M.S., Notes on exceptional Lie symmetry groups hierarchy and possible implications for E-infinity high energy physics, Chaos, solitons & fractals, 35, 1, 67-70, (2008)
[7] El Naschie MS. Mohamed El Naschie answers a few questions about this month’s emerging research front in the field of physics. Thomason essential science indicators. http://esi-topics.com/erf/2004/october04-MohamedElNaschie.html.
[8] El Naschie, M.S., Hilbert, Fock and Cantorian spaces in the quantum two-slit gedanken experiment, Chaos, solitons & fractals, 27, 1, 39-42, (2006) · Zbl 1082.81502
[9] El Naschie, M.S., A review of E-infinity theory and the mass spectrum of high energy particle physics, Chaos, solitons & fractals, 19, 1, 209-236, (2004) · Zbl 1071.81501
[10] El Naschie, M.S., Hilbert space, the number of Higgs particles and the quantum two-slit experiment, Chaos, solitons & fractals, 27, 1, 9-13, (2006) · Zbl 1082.81501
[11] Iovane, G.; Mohamed, E.I., Naschies \(\operatorname{\&z.epsiv;}^\infty\) Cantorian spacectime and its consequences in cosmology, Chaos, solitons & fractals, 25, 3, 775-779, (2005)
[12] Iovane, G., Waveguiding and mirroring effects in stochastic self-similar and fractal universe, Chaos, solitons & fractals, 23, 3, 691-700, (2004) · Zbl 1070.83542
[13] Iovane, G.; Giordano, P., Wavelets and multiresolution analysis: nature of \(\operatorname{\&z.epsiv;}^{(\infty)}\) Cantorian space-time, Chaos, solitons & fractals, 32, 3, 896-910, (2007)
[14] Masry, E., The wavelet transform of stochastic processes with stationary increments and its application to fractional Brownian motion, IEEE trans inform theory, 39, 1, 260-264, (1993) · Zbl 0768.60036
[15] Duffin, R.J.; Schaeffer, A.C., A class of nonharmonic Fourier series, Trans am math soc, 72, 2, 341-366, (1952) · Zbl 0049.32401
[16] Daubechies, I., The wavelet transform, time-frequency localization and signal analysis, IEEE trans inform theory, 36, 9, 961-1005, (1990) · Zbl 0738.94004
[17] Daubechies I. Ten lectures on wavelets. In: CBS-NSF regional conferences in applied mathematics, vol. 61. SIAM; 1992. MR 93e:42045. · Zbl 0776.42018
[18] Bownik, M., The structure of shift-invariant subspace of \(L^2(R^d)\),, J funct anal, 177, 2, 282-309, (2000) · Zbl 0986.46018
[19] Baggett, L.; Medina, H.; Merrill, K., Generalized mulitiresolution analyses, and a construction procedure for all wavelet sets in \(R^d\), J, Fourier anal appl, 5, 6, 563-573, (1999) · Zbl 0972.42021
[20] Bownik, M.; Rzeszotnik, Z.; Speegle, D., A characterization of dimension function of orthogonal wavelets, Appl comput harmon anal, 10, 1, 79-92, (2001) · Zbl 0979.42018
[21] Bakić, D., Semi-orthogonal Parseval frame wavelets and generalized multiresolution analyses, Appl comput harmon anal, 21, 3, 281-304, (2006) · Zbl 1106.42026
[22] Liu, Z.; Hu, G.; Wu, G., Frame scaling function sets and frame wavelet sets in \(R^d\), Chaos, solitons & fractals, 40, 5, 2483-2490, (2009) · Zbl 1198.42049
[23] Liu, Z.; Hu, G.; Wu, G.; Jiang, B., Semi-orthogonal frame wavelets and Parseval frame wavelets associated with GMRA, Chaos, solitons & fractals, 38, 5, 1449-1456, (2008) · Zbl 1198.42050
[24] Dai, X.; Larson, D., Wandering vectors for unitary systems and orthogonal wavelets, Mem AMS, 134, 640, (1998)
[25] Hernańde, E.; Weiss, G., A first course on wavelets, (1996), CRC Press Boca Raton, F1
[26] Bakić, D.; Krishtal, I.; Wilson, E.N., Parseval frame wavelets with \(E_n^{(2)}\)-dilations, Appl comput harmon anal, 19, 3, 386-431, (2005) · Zbl 1090.42020
[27] Dai, X.; Diao, Y.; Gu, Q.; Han, D., Frame wavelet sets in \(R^d\), J comput appl math, 155, 1, 69-82, (2003) · Zbl 1021.42019
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