Abstract
This chapter presents abstract constructions of GMRAs and wavelets using direct sums and direct limits. As a first example, we take up super-wavelets, which were developed in direct sum spaces to handle the application of multiplexing, the sending of multiple signals on a carrier at the same time. Direct limits, the second construction technique we discuss, were first used to build classical MRA’s and wavelets from filters, and later generalized to construct abstract GMRA’s from multiplicity functions. Finally we present a technique that uses direct sums to find a classifying set for all GMRA’s with finite multiplicity function and Haar measure class.
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References
Baggett, L., Courter, J., Merrill, K.: The construction of wavelets from generalized conjugate mirror filters in \(L^2(\mathbb R^n)\). Appl. Comput. Harmon. Anal. 13, 201–223 (2002)
Baggett, L., Furst, V., Merrill, K., Packer, J.: Generalized filters, the low-pass condition, and connections to multiresolution analysis. J. Funct. Anal. 257, 2760–2779 (2009)
Baggett, L., Larsen, N., Merrill, K., Packer, J., Raeburn, I.: Generalized multiresolution analyses with given multiplicity functions. J. Fourier Anal. Appl. 15, 616–633 (2009)
Baggett, L., Furst, V., Merrill, K., Packer, J.: Classification of generalized multiresolution analyses. J. Funct. Anal. 258, 4210–4228 (2010)
Baggett, L., Larsen, N., Packer, J., Raeburn, I., Ramsay, A.: Direct limits, multiresolution analyses, and wavelets. J. Funct. Anal. 258, 2714–2738 (2010)
Beals, R.: Operators in functions spaces which commute with multiplications. Duke Math. J. 35, 353–362 (1968)
Bildea, S., Dutkay, D., Picioroaga, G.: MRA super-wavelets. New York J. Math. 11, 1–19 (2005)
Courter, J.: Construction of dilation d wavelets. Contemp. Math. 247, 183–206 (1999)
Dutkay, D.: The wavelet Galerkin operator. J. Oper. Theory 51, 49–70 (2004)
Dutkay, D.: Low-pass filters and representations of the Baumslag-Solitar group. Trans. Am. Math. Soc. 358, 5271–5291 (2006)
Dutkay, D., Jorgensen, P.: Wavelets on fractals. Rev. Mat. Iberoamericana 22, 131–180 (2006)
Dutkay, D. Jorgensen, P.: Oversampling generates super-wavelets. Proc. Am. Math. Soc. 135, 2219–2227 (2007)
Dutkay, D., Jorgensen, P.: Fourier series on fractals: a parallel with wavelet theory. Contemp. Math. 464, 76–101 (2008)
Han, D., Larson, D.: Frames, bases and group representations. Mem. Am. Math. Soc. 147, x+94 (2000)
Larsen, N., Raeburn, I.: From filters to wavelets via direct limits. Contemp. Math. 414, 35–40 (2006)
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Merrill, K.D. (2018). Abstract Constructions of GMRAs. In: Generalized Multiresolution Analyses. Applied and Numerical Harmonic Analysis(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-99175-7_8
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DOI: https://doi.org/10.1007/978-3-319-99175-7_8
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