The Time operator of wavelets I.E. Antoniou a, * , K.E. Gustafson b,c a International Solvay Institute for Physics and Chemistry, and Theoretische Natuurkunde, University of Brussels, CP 231, 1050 Brussels, Belgium b Department of Mathematics, University of Colorado, Boulder, CO 80309±0395, USA c International Solvay Institute for Physics and Chemistry, University of Brussels, 1050 Brussels, Belgium Abstract This paper establishes an interesting new and general connection between the wavelet theory of harmonic analysis and the Time operator theory of statistical physics. In particular, it will be shown that an arbitrary wavelet multiresolution analysis (MRA) de®nes a Time operator T whose age eigenspaces are the wavelet detail subspaces W n . Extension of this result to the continuous parameter case induces a new notion of continuous wavelet multiresolution analysis. The Time operator T incorporates and exhibits in a natural way all ®ve fundamental properties of a wavelet multiresolution analysis. Ó 1999 Elsevier Science Ltd. All rights reserved. 1. Introduction Wavelets are known to have intimate connections to several other parts of mathematics, notably, phase space analysis of signal processing, reproducing kernel Hilbert spaces, coherent states in quantum me- chanics, spline approximation theory, windowed Fourier transforms and ®lter banks. There is already a large literature on wavelets, and on the connections mentioned above to other parts of mathematics and physics. Three excellent recent books Daubechies [1], Chui [2], Meyer [3], among others, report on these developments and connections, and will supply the reader with a wide variety of references. See also the survey by Heil and Walnut [4], and the expos es by Strang [5,6]. Briggs and Henson [7] examine similarities between wavelet multiresolutions and multigrid methods. Strang and Njuyen [8] examine relations between wavelets and ®lter banks. The notion of multiresolution analysis (MRA) in wavelet theory, which has become the essential framework within which to understand and explore wavelet structures, can be traced to the work of Meyer [9] and Mallat [10]. Approximation theorists at about the same time were considering related subdivision schemes, see Chui [11]. The earlier Laplacian pyramid algorithm due to Burt and Adelson [12] and subband ®ltering schemes have properties of a multiresolution analysis. For a full review of the wavelet theory in terms of wavelet multiresolution analysis, see the recent survey by Jawerth and Sweldens [13]. From the properties of a wavelet multiresolution analysis, here we are able to establish a new connection to another part of mathematics, speci®cally to the Time operator of statistical physics. First, a new characterization of the wavelet subspace W 0 of the MRA as the wandering generating subspace of the scaling transformation of the wavelet is established. Although this characterization of the wavelet subspace W 0 as a generating wandering subspace is natural, we have not seen it elsewhere. This new development of wavelet scaling in terms of the wandering subspace theory is given in Section 2. It may be regarded as a new operator-theoretic characterization of the scaling transformation of wavelets. Chaos, Solitons and Fractals 11 (2000) 443±452 www.elsevier.nl/locate/chaos * Corresponding author. E-mail address: antoniou@solvayins.ulb.ac.be (I.E. Antoniou) 0960-0779/99/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 8 ) 0 0 3 1 2 - 9