Abstract
This article provides an expository review of the relationship between graph \(C^{*}\)-algebras and the wavelet theory. This connection was first established by O. Bratteli and P.E.T. Jorgensen in the late \(90'\)s. Various mathematicians paid attention to this connection in the past 25 years, and the area has become active and dynamic now. The major contributors include O. Bratteli, P. E. T. Jorgensen, C. Farsi, E. Gillaspy, S. Kang, J. Packer, M. Marcolli and A. M. Paolucci. Our aim is to provide an expository survey in this direction.
This is a preview of subscription content, access via your institution.
References
T. Bates, D. Pask, I. Raeburn, and W. Szymanski, The\({C}^*\)-algebras of row-finite graphs, New York J. Math 6 (2000), no. 307, 324.
S. Bezuglyi and P.E.T. Jorgensen, Representations of Cuntz-Krieger relations, dynamics on bratteli diagrams, and path-space measures, (2015), 57–88.
A. Boggess and F. J. Narcowich, A first course in wavelets with Fourier analysis, second ed., John Wiley & Sons, Inc., Hoboken, NJ, 2009.
O. Bratteli, D.E. Evans, and P.E.T. Jorgensen, Compactly supported wavelets and representations of the Cuntz relations, Appl. Comput. Harmon. Anal. 8 (2000), no. 2, 166–196.
O. Bratteli and P. E. T. Jorgensen, Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale\(N\), Integral Equations and Operator Theory 28 (1997), no. 4, 382–443.
O. Bratteli and P.E.T. Jorgenson, A connection between multiresolution wavelet theory of scale n and representations of the Cuntz algebra\(o_n\), (1996).
O. Bratteli and P.E.T. Jorgenson, Iterated function systems and permutation representations of the Cuntz algebra, 1996.
O. Bratteli and P.E.T. Jorgenson, Wavelets through a looking glass: The world of the spectrum, Springer Science & Business Media, 2013.
M. Crovella and E. Kolaczyk, Graph wavelets for spatial traffic analysis, IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No. 03CH37428), vol. 3, IEEE, 2003, pp. 1848–1857.
I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
P.Y. Dibal, E. N. Onwuka, J. Agajo, and C. O. Alenoghena, Analysis of wavelet transform design via filter bank technique, (2019).
D. E. Dutkay, J. Haussermann, and P.E.T. Jorgensen, Atomic representations of Cuntz algebras, J. Math. Anal. Appl. 421 (2015), no. 1, 215–243.
D. E. Dutkay, G. Picioroaga, and M.S. Song, Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl. 409 (2014), no. 2, 1128–1139.
D.E. Dutkay and P.E.T. Jorgenson, Wavelets on fractals, Revista Matematica Iberoamericana 22 (2003), 131–180.
D.E. Dutkay and P.E.T. Jorgenson, Monic representations of the Cuntz algebra and Markov measures, Journal of Functional Analysis 4 (2014), no. 267, 1011–1034.
M. Enomoto, A graph theory for\({C}^*\)-algebras, Math. Japon. 25 (1980), 435–442.
C. Farsi, E. Gillapsy, S. Kang and J. Packer, Wavelets and graph\({C}^*\)-algebras, Excursions in Harmonic Analysis, Volume 5, Springer, 2017, pp. 35–86.
M. W. Frazier, An introduction to wavelets through linear algebra, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1999.
D. Goncalves, H. Li, and D. Royer, Branching systems for higher-rank graph\({C}^*\)-algebras, Glasg. Math. J. 60 (2018), no. 3, 731–751.
R.C. Guido, A note on a practical relationship between filter coefficients and scaling and wavelet functions of discrete wavelet transforms, Applied Mathematics Letters 24 (2011), no. 7, 1257–1259.
E.C. Jeong, Irreducible representations of the Cuntz algebra\( o_n\), Proc. Amer. Math. Soc. 127 (1999), no. 12, 3583–3590.
A. Jonsson, Wavelets on fractals and Besov spaces, J. Fourier Anal. Appl. 4 (1998), no. 3, 329–340.
P.E.T. Jorgensen, Minimality of the data in wavelet filters, Adv. Math. 159 (2001), no. 2, 143–228, With an appendix by Brian Treadway.
P.E.T. Jorgensen, Use of operator algebras in the analysis of measures from wavelets and iterated function systems, (2005), 13–26.
K. Kawamura, The Perron-Frobenius operators, invariant measures and representations of the Cuntz-Krieger algebras, Journal of Mathematical Physics 46 (2005), 083514.
A. Kumjian, D. Pask, and I. Raeburn, Cuntz–Krieger algebras of directed graphs, pacific journal of mathematics 184 (1998), no. 1, 161–174.
S. G. Mallat, Multiresolution approximations and wavelet orthonormal bases of\(L^2(\mathbb{R})\), Transactions of the American mathematical society 315 (1989), no. 1, 69–87.
M. Marcolli and A. M. Paolucci, Cuntz-Krieger algebras and wavelets on fractals, Complex Analysis and Operator Theory 5 (2011), no. 1, 41–81.
Gerard J. Murphy, \(C^*\)-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990.
I. Raeburn, Graph algebras, no. 103, American Mathematical Soc., 2005.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jaydeb Sarkar.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Anjali, V.A., Kumar, V.B.K. & Shankar, P. An Expository Note on Wavelets and Graph \(C^{*}\)-algebras. Indian J Pure Appl Math (2022). https://doi.org/10.1007/s13226-022-00351-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13226-022-00351-5
Keywords
- Graph \(C^{*}\)-algebras
- Wavelets
- Multiresolution Analysis
Mathematics Subject Classification
- Primary
- Secondary