Abstract
We analyze the structure of co-invariant subspaces for representations of the Cuntz algebras \({\mathcal{O}_N} \) for N = 2,3 ... , N < ∞ with special attention to the representations which are associated to orthonormal and tight-frame wavelets in L2 (ℝ) corresponding to scale number N..
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L.W. Baggett and K.D. Merrill, Abstract harmonic analysis and wavelets in Rn, The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, 1999) (L.W. Baggett and D.R. Larson, eds.), Contemp. Math., vol. 247, American Mathematical Society, Providence, 1999, pp. 17–27.
V. Baladi, Positive Transfer Operators and Decay of Correlations World Scientific, River Edge, NJ, Singapore, 2000.
J.A. Ball and V. Vinnikov, Functional models for representations of the Cuntz algebra preprint, 2002, Virginia Polytechnic Institute and State University.
J.A. Ball and V. Vinnikov, Lax-Phillips scattering and conservative linear systems: A Cuntz-algebra multi-dimensional setting preprint, 2002, Virginia Polytechnic Institute and State University.
N. Bourbaki, General Topology Elements of Mathematics, Springer, 1989, Chapters 1–2 (translated from the French).
O. Bratteli and P.E.T. Jorgensen, Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N Integral Equations Operator Theory 28 (1997), 382–443.
O. Bratteli and P.E.T. Jorgensen, Convergence of the cascade algorithm at irregular scaling functions The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, 1999) (L.W. Baggett and D.R. Larson, eds.), Con-temp. Math., vol. 247, American Mathematical Society, Providence, 1999, pp. 93–130.
O. Bratteli and P.E.T. Jorgensen, Wavelets through a Looking Glass: The World of the Spectrum Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, 2002.
O. Bratteli, P.E.T. Jorgensen, and V. Ostrovs’kyi, Representation theory and numerical AF-invariants: The representations and centralizers of certain states on Od Mem. Amer. Math. Soc. 168, No. 797.
J. Cuntz, Simple C* -algebras generated by isometries Comm. Math. Phys. 57 (1977), 173–185.
X. Dai and D.R. Larson, Wandering vectors for unitary systems and orthogonal wavelets Mem. Amer. Math. Soc. 134 (1998), no. 640.
I. Daubechies, Ten Lectures on Wavelets CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, SIAM, Philadelphia, 1992.
K.R. Davidson, D.W. Kribs, and M.E. Shpigel, Isometric dilations of non- commuting finite rank n-tuples Canad. J. Math. 53 (2001), 506–545.
N. Dunford and J.T. Schwartz, Linear Operators Interscience Publishers, Inc., New York, 1958–1971.
Ultrafilter Encyclopaedia of Mathematics, vol. 9 (M. Hazewinkel, ed.), Kluwer Academic Publishers, Dordrecht, Boston, London, 1993, pp. 301–302.
H. Helson, Lectures on Invariant Subspaces, Academic Press, New York, 1964.
C.T. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classesd’opérations non complètement continues Ann. of Math. (2) 52 (1950), 140–147.
P.E.T. Jorgensen, A geometric approach to the cascade approximation operator for wavelets Integral Equations Operator Theory 35 (1999), 125–171.
P.E.T. Jorgensen, Compactly supported wavelets and representations of the Cuntz relations, II Wavelet Applications in Signal and Image Processing VIII (San Diego, 2000) (A. Aldroubi, A.F. Laine, and M.A. Unser, eds.), Proceedings of SPIE, vol. 4119, SPIE, Bellingham, WA, 2000, pp. 346–355.
P.E.T. Jorgensen, Minimality of the data in wavelet filters, Adv. Math. 159 (2001), 143–228.
P.E.T. Jorgensen and D. Kribs, Wavelet representations and Fock space on positive matrices J. Funct. Anal. 197 (2003), 526–559.
P.E.T. Jorgensen, L.M. Schmitt, and R.F. Werner, Positive representations of general commutation relations allowing Wick ordering J. Funct. Anal. 134 (1995), 33–99.
D. Kribs, Quantum channels, wavelets, dilations, and representations of ON Proc. Edinburgh Math. Soc., to appear.
S.G. Mallat, A Wavelet Tour of Signal Processing 2nd ed., Academic Press, Orlando - San Diego, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Basel AG
About this paper
Cite this paper
Jorgensen, P.E.T. (2004). Closed Subspaces which are Attractors for Representations of the Cuntz Algebras. In: Ball, J.A., Helton, J.W., Klaus, M., Rodman, L. (eds) Current Trends in Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 149. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7881-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7881-4_11
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9608-5
Online ISBN: 978-3-0348-7881-4
eBook Packages: Springer Book Archive