Abstract
In this paper we show how wavelets originating from multiresolution analysis of scaleN give rise to certain representations of the Cuntz algebrasO N , and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space\(L^2 (\mathbb{T})\) by (S i ξ) (z)=m i (z)ξ(z N). We characterize the Wold decomposition of such operators. If the operators come from wavelets they are shifts, and this can be used to realize the representation on a certain Hardy space over\(L^2 (\mathbb{T})\). This is used to compare the usual scale-2 theory of wavelets with the scale-N theory. Also some other representations ofO N of the above form called diagonal representations are characterized and classified up to unitary equivalence by a homological invariant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[Ada62] J.F. Adams,Vector fields on spheres, Ann. of Math. (2)75 (1962), 603–632.
[BaRu96] V. Baladi and D. Ruelle,Sharp determinants, Invent. Math.123 (1996), 553–574.
[BJP96] Ola Bratteli, Palle E.T. Jorgensen, and Geoffrey L. Price,Endomorphisms of B(H), Quantization, nonlinear partial differential equations, and operator algebra (William Arveson, Thomas Branson, and Irving Segal, eds.), Proc. Sympos. Pure Math., vol. 59, American Mathematical Society, 1996, pp. 93–138.
[BrJo96a] Ola Bratteli and Palle E.T. Jorgensen,Endomorphisms of B(H), II: Finitely correlated states on B(H), J. Funct. Anal., to appear.
[BrJo96b] Ola Bratteli and Palle E.T. Jorgensen,Iterated function systems and permutation representations of the Cuntz algebra, submitted to Mem. Amer. Math. Soc., 1996.
[Bro65] Hans Brolin,Invariant sets under iteration of rational functions., Ark. Mat.6 (1965), 103–144.
[BrRo96] Ola Bratteli and Derek W. Robinson,Operator algebras and quantum statistical mechanics, vol. II, 2nd ed., Springer-Verlag, Berlin-New York, 1996.
[CaGa93] Lennart Carleson and Theodore W. Gamelin,Complex dynamics, Springer-Verlag, New York, 1993.
[CFS82] I.P. Cornfeld, S.V. Fomin, and Ya. G. Sinai,Ergodic theory, Grundlehren der mathematischen Wissenschaften, vol. 245, Springer-Verlag, New York, 1982, translated from the Russian by A.B. Sosinskii.
[CoDa96] Albert Cohen and Ingrid Daubechies,A new technique to estimate the regularity of refinable functions, Rev. Mat. Iberoamericana12 (1996), 527–591.
[Coh90] A. Cohen,Ondelettes, analyses multirésolutions et filtres miroirs en quadrature, Ann. Inst. H. Poincaré Anal. Non Lineaire7 (1990), 439–459.
[CoMa95] Carl C. Cowen and Barbara D. MacCluer,Composition operators on spaces of analytic functions, CRC Press, Boca Raton, FL, 1995.
[CoRy95] Albert Cohen and Robert D. Ryan,Wavelets and multiscale signal processing, Applied Mathematics and Mathematical Computation, vol. 11, Chapman & Hall, London, 1995, translated from the French by Ryan; revised version of Cohen's doctoral thesisOndelettes et traitement numérique du signal, Masson, Paris, 1992.
[Cun77] Joachim Cuntz,Simple C *-algebras generated by isometries, Comm. Math. Phys.57 (1977), 173–185.
[DaPi96] Ken R. Davidson and David R. Pitts,Free semigroup algebras, preprint, University of Waterloo, 1996.
[Dau92] Ingrid Daubechies,Ten lectures on wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, Society for Industrial and Applied Mathematics, Philadelphia, 1992.
[GrHa94] Karlheinz Gröchenig and Andrew Haas,Self-similar lattice tilings, J. Fourier Anal. Appl.1 (1994), 131–170.
[GrMa92] Karlheinz Gröchenig and W.R. Madych,Multiresolution analysis, Haar bases, and self-similar tilings of R n, IEEE Trans. Inform. Theory38 (1992), 556–568.
[Ho96] Mark C. Ho,Properties of slant Toeplitz operators, preprint, Purdue University, 1996.
[HoJa96] William E. Hornor and James E. Jamison,Weighted composition operators on Hilbert spaces of vector-valued functions, Proc. Amer. Math. Soc.124 (1996), 3123–3130.
[Hut81] John E. Hutchinson,Fractals and self-similarity, Indiana Univ. Math. J.30 (1981), 713–747.
[JoPe94] Palle E.T. Jorgensen and Steen Pedersen,Harmonic analysis and fractal limitmeasures induced by representations of a certain C *-algebra, J. Funct. Anal.125 (1994), 90–110.
[JoPe96] —,Harmonic analysis of fractal measures, Constr. Approx.12 (1996), 1–30.
[Jor95] Palle E.T. Jorgensen,Harmonic analysis of fractal processes via C *-algebras, preprint, The University of Iowa, 1995.
[Kea72] Michael Keane,Strongly mixing g-measures, Invent. Math.16 (1972), 309–324.
[Lam86] Alan Lambert,Hyponormal composition operators, Bull. London Math. Soc.18 (1986), 395–400.
[LaPh89] Peter D. Lax and Ralph S. Phillips,Scattering theory, 2nd ed., Pure and Applied Mathematics, vol. 26, Academic Press, Inc., Boston, 1989, with appendices by Cathleen S. Morawetz and Georg Schmidt.
[LaSt91] Yu. D. Latushkin and A. M. Stëpin,Weighted shift operators and linear extensions of dynamical systems, Russian Math. Surveys46 (1991), 95–165, translation ofAkademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo, Uspekhi Matematicheskikh Nauk 46 (1991), 85–143, 240.
[Mal89] Stephane G. Mallat,Multiresolution approximations and wavelet orthonormal bases of L 2(R), Trans. Amer. Math. Soc.315 (1989), 69–87.
[MePa93] Yves Meyer and Freddy Paiva,Remarques sur la construction des ondelettes orthogonales, J. Anal. Math.60 (1993), 227–240.
[Mey87] Yves Meyer,Ondelettes, fonctions splines et analyses graduées, Rend. Sem. Mat. Univ. Politec. Torino45 (1987), 1–42.
[Mey92] —,Ondelettes et algorithmes concurrents, Actualité Scientifiques et Industrielles, vol. 1435, Hermann, Paris, 1992.
[Mey93] —,Wavelets: Algorithms & applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1993, translated from the French and with a foreword by Robert D. Ryan.
[MRF96] Peter R. Massopust, David K. Ruch, and Patrick J. Van Fleet,On the support properties of scaling vectors, Appl. Comput. Harmon. Anal.3 (1996), 229–238.
[PaPo90] William Parry and Mark Pollicott,Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, vol. 187–188, Société Mathématique de France, Paris, 1990.
[Pow88] Robert T. Powers,An index theory for semigroups of *-endomorphisms of B(H) and type II1 factors, Canad. J. Math.40 (1988), 86–114.
[Rue94] D. Ruelle,Dynamical zeta functions for piecewise monotone maps of the interval, CRM Monograph Series, vol. 4, American Mathematical Society, Providence, 1994.
[SzFo70] Béla Szökefalvi-Nagy and Ciprian Foias,Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam-London, Elsevier, New York, Akadémiai Kiadó, Budapest, 1970, translation and revised edition of the original French edition, Masson et Cie, Paris, Akadémiai Kiadó, Budapest, 1967.
[Wal82] Peter Walters,An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York, 1982.
[Wal96] —,Topological Wiener-Wintner ergodic theorems and a random L 2 ergodic theorem, Ergodic Theory Dynam. Systems16 (1996), 179–206.
Author information
Authors and Affiliations
Additional information
Work supproted in part by the U.S. National Science Foundation and the Norwegian Research Council.
Rights and permissions
About this article
Cite this article
Bratteli, O., Jorgensen, P.E.T. Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scaleN . Integr equ oper theory 28, 382–443 (1997). https://doi.org/10.1007/BF01309155
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01309155