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The Time operator of wavelets
Chaos, Solitons and Fractals 11 (2000) 443±452
www.elsevier.nl/locate/chaos
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 Wn . 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 mechanics, 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 exposes 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 W0 of the MRA as the wandering generating subspace of the scaling transformation of the wavelet is established. Although this characterization of the wavelet subspace W0 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. *
Corresponding author. E-mail address: [email protected] (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
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I.E. Antoniou, K.E. Gustafson / Chaos, Solitons and Fractals 11 (2000) 443±452
In Section 3 we construct the Time operator of an arbitrary wavelet multiresolution analysis. This result is obtained by using the wandering subspace characterization obtained in Section 2 in conjunction with discrete systems of imprimitivity through which the Time operator of statistical physics is de®ned. The Time operator is seen to incorporate and exhibit in a natural way all ®ve fundamental properties of a wavelet multiresolution analysis. Then in Section 4 we address the question of extending the construction of Section 3 to a Time operator with continuous parameter. Time operators with continuous parameter occur naturally in statistical physics and quantum mechanics. Such Time operators may be viewed as playing a role analogous to that of a position operator Q in a Schr odinger couple. Therefore in Section 4 we present a new notion of continuous multiresolution analysis as naturally induced by the construction of a corresponding Time operator. In Section 5 we observe that the Haar wavelet subspaces Wn are now most naturally seen as the eigenspaces of their Time operator which by our construction is conjugate to the wavelet scaling transformations in the sense of the canonical commutation relations. From this new viewpoint the Haar basis may now be regarded most naturally as the basis of its corresponding selfadjoint Time operator, and thus as an eigenbasis whose role is to be canonically conjugate to its scaling. The same statement applies to any wavelet basis, but we highlight the Haar basis because Haar's original tasking [14,15] was to ®nd a complete orthonormal set which was not an eigenbasis derived naturally from some selfadjoint Sturm±Liouville dierential operator. Section 6 discusses our conclusions. In this paper we stay completely in L2R but extensions to higher dimensions in principle may be realized. Part of our motivation for the continuous parameter case came from an earlier work [16] in which we connected regular stationary stochastic processes to the position and momentum operators of quantum mechanical evolutions. Our motivation in the discrete parameter case came from an earlier work [17] relating Kolmogorov automorphisms and Haar Systems. We have known now for a number of years that we could obtain Time operators for wavelet structures. A preliminary announcement has been given in our lectures [18]. But it was not until we found the precise characterization given here of wavelet scaling in terms of the wandering subspace theory in the discrete parameter case, and not until we could clarify the appropriate notion of continuous wavelet multiresolution analysis given here for the continous parameter case, that we felt we had a full understanding of the essentials of this new connection of wavelets to the Time operator of statistical physics.
2. Wavelets as wandering subspsaces A discrete MRA [1±3,13] of square integrable functions over the real line consists of a hierarchy of approximations de®ned as averages on dierent scales. The ®ner the scale the better the approximation. If Pn is the orthoprojection corresponding to the approximation at a resolution 2n , the ranges Hn , n 0; 1; 2; . . . of Pn de®ne a sequence of Hilbert subspaces of L2R with the properties of MRA: (1) H T n Hn1 (2) Sn Hn f0g (3) n Hn is dense in L2R (4) A function f x is in the space Hn if and only if the scaled function f 2x is in the space Hn1 . (5) There exists a function / in H0 such that the set of translated functions /a x / x ÿ a, a 0; 1; 2; . . . is a orthonormal basis for the space H0 . We remark that this last property (5) of an MRA is stated in a number of ways in the literature, but all versions either explicitly or implicitly imply that such a function / has been chosen or can be constructed. For example it is sucient to require that one can ®nd a function g x in L2R such that the translated functions g x ÿ a form a Riesz basis of H0 . Then one can construct an orthonormal basis / x ÿ a for H0 . From the scale function / one can construct in standard ways [1±3] an oscillatory wavelet w in L2R such that the discrete translations and scalings of w provide orthonormal bases in L2R . We will not consider here the generalizations in which there are several generating functions /i , nor will we consider the multidimensional setting L2 Rn , although the new connections to dynamical systems which we establish in this paper could surely be pursued in those more general contexts.
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The successive multiresolution approximations of a function f are expressed through the projections Pn onto Hn : X Pn f h/na ; f i/n;a ; 1 a2Z
R 1 where hg; f i denotes the inner product ÿ1 dxg xf x. The detail or new information between two successive approximations Pn f and Pn1 f of the functions f is given by the dierence Pn1 ÿ Pn f . The range of the projection Pn1 ÿ Pn is the orthocomplement Wn of the space Hn in Hn1 Wn Hn1 Hn ;
Wn ? Hn :
2
The properties (1±5) of the MRA imply that the detail spaces Wn are also the scalings of W0 and that they are mutually orthogonal spaces generating all the space L2R . We now wish to reformulate the meaning of scaling within a wavelet multiresolution analysis. Let us recall ®rst a few basic facts. Condition (5) of the wavelet multiresolution analysis means that the function / is a cyclic vector for unitary representation of the discrete translation group Ua / x / x ÿ a, a 2 Z, on the space H0 . As the spaces Hn are just scalings of the space H0 (Condition 4), each scaled function / 2n x is also a cyclic vector for the unitary representation of the discrete translation group on the space Hn , n 2 Z. The function / gives rise therefore to the orthonormal basis /na x 2n=2 / 2n x ÿ a;
a2Z
3
of the space Hn . Next we recall the unitary representation of the ane group of translations and scalings, namely, 1 1 ÿ1 Uab f x p f bÿ1 x ÿ a p f a; b x; jbj jbj
4
where a 2 R, b 2 R ÿ f0g. Recall that the ane group R R ÿ f0g acts on the real line as follows: x7! a; bx b x a
5
i.e., ®rst translation by a 2 R and then scaling by b 6 0. The group synthesis is 0
a; b a0 ; b0 bb0 ; a0 b ÿ1 a:
6
The unit and inverse transformations are I 0; 1; a; b
ÿ1
7
ÿba; bÿ1 :
8
Since the spaces Wn are obtained from each other by scalings and are mutually orthogonal, the functions wna x 2n=2 w 2n x ÿ a;
a2Z
9
for ®xed n 2 Z form an orthonormal basis for Wn . The set wna , n 2 Z, a 2 Z is therefore an orthonormal basis for the whole space L2R . We may also express the approximation projections Pn in terms of the wavelets wna X Pn1 ÿ Pn f hwna ; f iwna : 10 a2Z
From property (4) we see that the space Hn1 is the image of the space Hn under the unitary transformation p Vf x 2f 2x: 11 The operators V n provide a unitary representation on the L2R of the discrete scaling transformations pspace S : x7!2x. We remark that the transformation Vf x 2f 2x is justpthe Koopman operator [19,20] induced by the underlying point transformation S x 2x. The factor 2 is introduced in order to have a
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unitary representation. The properties (1±3) of the multiresolution analysis are the conditions for the unitary scaling transformation V to be a bilateral shift [21]. Such unitary shifts have absolutely continuous spectrum of uniform multiplicity [22]. Thus the wavelet space W W0 H1 H0 V H0 H0
12
is exactly the wandering generating subspace of the shift operator V. The term wandering subspace from the invariant subspace theory [21,22] means that V n W ? V m W;
n 6 m
13
and the term generating means that 1
L2 R V n W: nÿ1
14
The wandering generating subspace W W0 is clearly in®nite dimensional because the functions w x ÿ a, a 2 Z are an orthonormal basis for it. As the dimension of the wandering generating subspace W is the multiplicity of the shift [21,22], we conclude that the multiresolution analysis is a bilateral shift of in®nite countable multiplicity. Therefore we have proved Theorem 1. A family of projections Pn defines a MRA of the space L2R if and only if: (1) The ranges Hn of Pn are obtained from the wandering generating subspace W of the bilateral shift V corresponding to the scaling transformation x 7! 2x through the formula: n
Hn V k W: kÿ1
15
(2) The wandering generating subspace W is a cyclic subspace for the group of translations. Each family of cyclic vectors for the translation group is a family of wavelets generating the multiresolution Pn n2Z . The multiplicity of the shift is equal to the number of linearly independent wavelets generating the multiresolution Pn n2Z . Thus a wavelet multiresolution analysis (1)±(5) is a bilateral shift of scalings on L2R with countable in®nite multiplicity such that the wandering generating subspace of the scaling shift operator is also a cyclic representation space with respect to the group of discrete translation. The cyclic vectors are the wavelets. In the next section we will employ this operator-theoretic characterization of the scaling transformation of wavelets as a part of imprimitivity relations which will de®ne a Time operator for an arbitrary wavelet MRA. Remark. Daubechies ([1], p. 129), points out that ``there are many `ladders' of spaces which satisfy the wavelet MRA properties (1)±(3),'' and that it is the additional requirement (4) that ``f 2 Vj () f 2j 2 V 0 '' that is the key requirement of multiresolution analysis. Theorem 1 may be regarded as the deeper content of that statement. In fact we believe that statement and its concrete realization in Theorem 1 deserve even further elaboration. The essence of Property (4) of a multiresolution analysis is that the subspaces Hn of Properties (1)±(3) be implemented as the exact increasing ranges of the unitary group V n . The multiplicity of H0 is transferred exactly to each Hn . The importance of the emphasis that we place upon this observation will become more apparent when we consider the continuous parameter case in Section 4. 3. The Time operator of wavelets The properties (1)±(3) of the multiresolution analysis imply that the approximation projections Pn satisfy the properties of a discrete resolution of the identity Pn , n 2 Z in the Hilbert space L2R , namely
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(1°) Pn < Pn1 , (2°) Pÿ1 limn!ÿ1 Pn 0, (3°) P1 limn!1 Pn I. It is also straightforward to see that property (4) of the multiresolution analysis when coupled with (1°)±(3°) means that Pn1 VPn V ÿ1 ;
16
where V is the unitary operator (11) associated to scalings and whose characterization is given in Theorem 1. The above properties (1°)±(3°), (16) are the properties of a discrete system of imprimitivity [23]. Therefore according to the de®nition of the Time operator [24±26,17] in statistical physics through such systems of imprimitivity, we are led to de®ne the Time operator of the multiresolution analysis by the formula X n Pn1 ÿ Pn : 17 T n2Z
In Eq. (17) T is a selfadjoint operator with dense domain D T . The imprimitivity conditions (1°)±(3°) (16) give the change of the Time operator at each stage n: V ÿn TV n T nI:
18
The multiresolution approximation projections Pn are the spectral projections of the Time operator T. From the wandering subspace relation (13) of the approximation projections Pn and the wavelets wna , we see that the action of the Time operator in terms of wavelets is X X n hwna ; f iwna x: 19 Tf x n2Z
a2Z
The relation of the wavelets with the Time operator is thus clear. The wavelet subspaces Wn are the age eigenspaces of the Time operator. Age n means resolution at the stage n. The analogy with statistical physics is striking because the higher the age the ®ner the detail. For any function f in its domain, the Time operator T attributes an average age which keeps step with the external time n according to hV n f ; TV n f i hf ; Tf i n:
20
Thus the following theorem has been proved. Theorem 2. Any wavelet MRA of the space L2 R defines a Time operator. The wavelet subspaces Wn of the MRA are the age eigenspaces of the Time operator. Note that the Time operator of Theorem 2 incorporates all ®ve MRA properties (1)±(5) in a natural way. Conditions (1)±(3) correspond to its spectral projections. Condition (4) corresponds to increasing time. Condition (5) by Theorem 1 couples the scaling wandering generating subspace W to the group of translations at each stage. Remark. An interesting way to view a role of the Time operator directly within the wavelet theory comes by asking the following question: given approximation projections (1°)±(3°), how may we characterize the wavelets w which are compatible with such given Pn ? We may introduce T Rn Pn1 ÿ Pn and show that any w compatible with the Pn defines a family wna of eigenfunctions of T such that Eq. (19) holds. Then any other wavelet w0 compatible with the Pn is characterized by w0na Xwna where X is unitary, XEn En X where En Pn1 ÿ Pn , and X commutes with translation. Remark. Time operators and their definition and meanings are intrinsically linked to the canonical commutation relations of quantum mechanics [24±26,16±18]. This will become more apparent in the next section where the continuous parameter wavelet case is considered. For discrete parameter as in the present section, the Time operator is defined in statistical physics [25,26] through a system of imprimitivity. In other words, the result of Theorem 2 is that wavelets determine their Time operator in exactly the same way as the Time operator occurs in statistical physics.
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4. Continuous wavelet multiresolution analysis In terms of the Time operator theory of statistical physics [16,17,24±26], it is natural to extend the results of Sections 2 and 3 to a continuous parameter. Historically, the continuous parameter wavelet theory was developed from the point of view of the wavelet transform. From that and the frame theory came discrete wavelet transforms which led to the concept of (discrete) multiresolution analysis [1±4]. Also, to our knowledge the wandering subspace theory essential to Theorems 1 and 2 has heretofore been restricted to the discrete parameter case. Thus, to our knowledge, our formulation of the concept of continuous wavelet multiresolution analysis introduced in this section for purposes of obtaining its Time operator for the continuous parameter case, is new. Our approach may be regarded in some sense as strongly reminiscent of the transition from discrete parameter Kolmogorov automorphisms to continuous parameter Kolmogorov ¯ows in the theory of stochastic processes [27]. In fact it was from this perspective [16,17] that we ®rst realized that one may obtain a Time operator for wavelet structures. We ®rst recall for the reader's convenience the following important result, which provides three formulations of the canonical commutation relations, which are fundamental to the foundation of quantum mechanics in terms of Heisenberg Algebras. The position and momentum operators in the Schr odinger representation on the space L2R have the form qf x xf x;
pf x ÿi
d x dx
21
and they satisfy the canonical commutation relation 1 p; q ÿI: i
22
Lemma 1 (CCR). Each of the following conditions is sufficient for two abstract selfadjoint operators Q; P on the Hilbert space H to be unitarily equivalent to the Schrodinger pair q; p on L2R , or to a direct sum of such pairs: 1 P ; Q ÿI i Z Q
kdPk ;
plus domain conditions;
23
Vt Pk Vt Pkt ;
24
Vt Us eÿits Us Vt ;
Vt eÿiPt ;
Vt eÿiPt ; Us eiQs :
25
The conditions (23)±(25) are often referred to as the Heisenberg, Schr odinger and Weyl, respectively, form of the canonical commutation relations. Lemma 1, often called the Stone±Von Neumann Theorem, has a long and rich history, with far-reaching implications and connections to many parts of mathematics and physics. For instance, the Weyl form may be regarded as a truncation of the Baker±Campbell± Hausdor theorem for the exponentiation of operator sums. The Schr odinger form is a version of imprimitivity systems from the theory of group representations. The Heisenberg form is the uncertainty relation from quantum mechanics. An example of the needed supplemental domain conditions in the Heisenberg form (Rellich±Dixmier): P 2 Q2 be essentially selfadjoint. For other conditions and further information, see Putnam [28]. The second result that we need to recall is key to the theory of Kolmogorov ¯ows in the ergodic theory [27] and also plays a key role in the scattering theory [29]. Lemma 2 (Von Neumann). Let there be given a strongly continuous one parameter group Vt in a Hilbert space H. Then the following conditions are equivalent.
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1. The group Vt has homogeneous Lebesgue spectrum:
449
26
2. There exists a subspace H0 of H such that i Ht Vt H0 H0 for t > 0: \ ii Vt H0 f0g:
27
ÿ1
iii
[
Vt H0 H:
ÿ1
3. There exists a strongly continuous unitary group Ut coupled to Vt through the commutation relations Vt Us eÿist Us Vt ;
ÿ1 < s; t < 1:
28
odinger couple in the sense of Lemma 2 implies that the in®nitesimal generators of Vt and Ut are a Schr Lemma 1. But Lemma 2 states more: the chain Ht is implemented by the group Vt . This will become the condition 4 of a continuous multiresolution analysis as we shall de®ne it. See our Remark at the end of Section 2 as concerns the importance of this observation. Let us now de®ne a continuous multiresolution analysis, motivated so that its properties yield a Time operator such as that obtained in Theorem 2 in the discrete parameter case. To that end we de®ne a continuous multiresolution analysis to be a continuous chain of Hilbert spaces Ht with the properties 10 H T t Hs for t < s, 20 St Ht f0g, 30 t Ht is dense in L2R , 40 Ht Vt H0 for some continuous unitary group Vt , 50 Vt is irreducible and has Lebesgue spectrum of countable multiplicity: Property (40 ) as we have stated embodies the key property of exact conjugate implementability but is otherwise permitted generality of group Vt . There are many examples of continuous MRA, notably in stochastic processes, where the increase of detail is that of r-algebra ®ltration. We have been able to show [30] in fact that all continuous multiresolution analyses may be strongly linked to such processes. It would be interesting in the future to investigate selected classes of continuous implementation groups Vt according to desired speci®c underlying wavelet group actions. In an intuitive sense [18] one such class would be a scaling action Vt f s cf bs, but technical details must be considered elsewhere. For any continuous multiresolution analysis (10 )±(50 ) we are now able to obtain a Time operator. Theorem 3. Any continuous multiresolution analysis defines a Time operator T. Proof. The (assumed right continuous) projection family Pt onto the subspaces Ht with properties (10 ), (20 ), (30 ), (50 ) is the spectral family of a selfadjoint operator with absolutely continuous Lebesgue spectrum of in®nite multiplicity. Exponentiation of that operator yields a strongly continuous unitary group Ut . By Lemma 2 the conditions (10 ), (20 ), (30 ), (40 ) guarantee the existence of a canonically conjugate continuous one parameter group Vs in the Weyl form (25,28) of the canonical commutation relations. It is straightforward to check [16,17], as in the discrete case, that the condition (40 ) means that Pkt Vt Pk Vt ; where Vt : H0 ! Ht is the unitary family of condition (40 ). Thus we are led to de®ne Z 1 T sdPs : ÿ1
29
30
By this construction, T is a ``position'' operator Q, position now being interpreted in the sense of time. For a function f in its domain D T , T attributes an average age
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hVt f ; TVt f i hf ; Tf i t
31
in which the age corresponds to the group action Vt which has been performed on f. This action or ``aging'' corresponds to the operator imprimitivity condition Vÿt TVt T tI
32
analogous to the discrete case (18). Note that the Time operator, now in the continuous parameter case, again naturally incorporates all ®ve properties 10 ± 50 of the continuous multiresolution analysis, in a meaningful way. Remark. One may obtain a more detailed representation of T in terms of cyclic vectors in the continuous parameter case, analogous to Eq. (19) in the discrete parameter case, by employing the more detailed operational calculus of unitarily equivalent selfadjoint operators with continuous spectrum [31]. See especially [31, Chs. VI, VII]. We find it especially amusing that the functions g; /; w of [31, pp. 244±250], play roles similar to those of the g; /; w of the wavelet MRA. It is of course due to the separability of L2R or H that one may obtain such spectral direct sum decompositions over the continuous spectrum. We placed an infinite multiplicity in condition (50 ) in analogy with the discrete MRA, but that need not necessarily be the case. A more detailed investigation of wavelet-like properties of such direct sum component cyclic vectors will be given elsewhere.
5. Haar's basis and the Time operator One of Haar's principal tasks in his dissertation [14] was to answer the question of whether there existed a complete orthonormal basis that was not the eigenbasis of any selfadjoint Sturm±Liouville dierential operator. This he answered armatively with his basis. This basis moreover serves as a monotone orthogonal basis for many Lp spaces [32]. With Theorem 2 we ®nd now the natural setting of Haar's basis, namely, as the eigenbasis of the selfadjoint Time operator. That is, the natural selfadjoint operator associated to the Haar basis, although not a Sturm±Liouville dierential operator, is just the Time operator of the Haar wavelets wna 2n=2 10;1 2n1 x ÿ 2a 2 ÿ 10;1 2n1 x ÿ 2a 1:
33
As we have pointed out earlier, the Time operator of any wavelet MRA may be viewed in the same way as representing the average aging by scaling of any given initial vector f. However, let us formalize this observation in terms of Haar's basis, due to its historical signi®cance as a basis constructed without regard to any given operator. Theorem 4. The natural setting of Haar's basis is that of eigenbasis of its associated Time operator. The natural role of Haar's basis is that of an eigenbasis canonically conjugate to its scaling. Remark. As Haar's basis, by his construction, becomes increasingly discontinuous with increasing n, that is the meaning of time in the Haar sense. Such a property is incompatible with needed regularity of any eigenbasis of any Sturm±Liouville boundary value problem. Remark. As we have pointed out elsewhere [15,17], Haar's three main contributions to mathematics were his generalization of Lebesgue measure to the construction of an invariant measure on any locally compact topological group, his construction of a complete orthonormal basis (33) not arriving from any regular Sturm± Liouville boundary value problem, and perhaps most overlooked, his proof that a wide class of Sturm±Liouville eigenfunction expansions converge or diverge pointwise along with the Fourier trigonometric expansion. We believe that Haar would be delighted to now see the Time operator as the correct and natural setting for his basis. Although the Time operator is not a selfadjoint second order dierential Sturm±Liouville operator, and indeed we would not want it to be such as that would violate Haar's motivation for constructing his basis,
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nonetheless the Time operator may be viewed as a ®rst order selfadjoint dierential operator, as follows. In the continuous parameter case, Section 4, by Fourier transform the Time operator may be seen to be a momentum operator p in a Schrodinger couple (21). However, as the considerations of Section 4 make clear, it is more natural to consider the Time operator as a position operator q. The same interpretation can be carried out in the discrete case, although the in®nitesimal generators are no longer p and q. The full details of this last statement must be worked out elsewhere. 6. Conclusions 1. A general connection between wavelet multiresolution analysis theory and the Time operator of statistical physics has been established. In particular, it was shown that any wavelet multiresolution analysis naturally de®nes a Time operator whose age eigenspaces are the wavelet detail subspaces Wn . 2. Intrinsic to establishing this fact is a new characterization of wavelet scaling in terms of wandering subspace theory. The generating wandering subspace is the wavelet subspace W0 . 3. Extending this Time operator construction for wavelets to the continuous parameter case led naturally to a new formulation of a continuous multiresolution analysis. In this formulation the essential property is that the wavelet chain be implemented exactly by a continuous unitary group. 4. We reinterpret Haar's basis, originally designed by him to be independent from any Sturm±Liouville dierential operator, as now naturally the eigenbasis of its corresponding wavelet Time operator. More generally, a natural interpretation of any wavelet basis is that of an eigenbasis canonically conjugate to its scaling. 5. We have learned from a recent extensive review [33] of coherent states and their generalizations that there has been concern about ``the missing link between the CWT based on the CS approach and the DWT derived from multiresolution'', [33]. An ``intermediate'' approach based upon wavelets on discrete ®elds has recently been oered [34]. We may suggest that our approach, based upon the Time operator theory of statistical physics and ergodic theory, oers a unifying view to the discrete and continuous wavelet theory. 6. Recent discussions with colleagues revealed to us two papers [35,36] which also deal, either implicitly [35] or explicitly [36], with a continuous multiresolution analysis. The ®rst paper [35] implicitly involves a notion of continuous MRA through the wavelet transform, in the earlier known sense of Calderon, Grossmann, Morlet. An interesting result [35] is that the wavelet scaling function can be expressed as a continuous in®nite product of dilated copies of a low-pass ®lter. However, it should be noted for comparison to our formulation 10 ± 50 of continuous MRA in Section IV that the work [35] does not include any such formulation. The second paper [36] is much closer to ours and does include a continuous MRA formulation: De®nition 3.1 of [36], properties (a) and (b), are essentially our 10 ± 30 . Note however that the chain Va of [36] is restricted to a the positive numbers. Properties (c) and (d) are essentially analogous to our (40 ) and (50 ), but stress instead (c) rescaling rather than (40 ) exact implementability of some continuous group Vt , and (d) translation invariance rather than (50 ) countable Lebesgue multiplicity. We originally [18] also thought in terms of the scaling group for property (40 ) but then, because our original motivation for the Time operator came from a similar construction in stochastic process theory [16], we decided that the essence of property 4 of wavelet MRAs was exact implementability by some group V n or Vt . We have commented on this point earlier in this paper. The limitations of restricting the continuous MRA to scaling groups have been investigated in [36]. In particular the generating analyzing wavelet may be nonunique and the scaling analysis approximations correspond to simple bandpass ®ltering. Thus our formulation would appear to be more general. Acknowledgements We would like to thank I. Prigogine for his interest and encouragement. This work was partly supported by the Belgian Interuniversity Attraction Poles, the NATO CR Grant 920977R and the European Commission ESPRIT Project 21042 CTIAC.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
I. Daubechies, Ten Lectures on Wavelets, SIAM Publications, Philadelphia, 1992. C. Chui, An Introduction to Wavelets, Academic Press, Boston, 1992. Y. Meyer, Wavelets and Operators, Cambridge Press, Cambridge, 1992. C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1989) 628. G. Strang, Wavelets and dilation equations: a brief introduction, SIAM Review 31 (1989) 614. G. Strang, Wavelets, American Scientist 82 (1994) 250. W. Briggs, V. Henson, Wavelets, multigrid, SIAM J. Sci. Comput. 14 (1993) 506. G. Strang, T. Nguyen, Wavelets and Filter Banks, Wesley±Cambridge Press, Boston, 1996. Y. Meyer, Ondelettes et fonctions splines, Seminaire EDP, Ecole Polytechnique, Paris, 1986. S. Mallat, Multiresolution Approximations and Wavelets, Ph.D. Dissertation, University of Pennsylvania, 1988. C. Chui, Multivariate Splines, SIAM Publications, Philadelphia, 1988. P. Burt, E. Adelson, The Laplacian Pyramid as a compact image code, IEEE Trans. Comm. 31 (1983) 482. B. Jawerth, W. Sweldens, An overview of wavelet based multiresolution analysis, SIAM Review 36 (1994) 377. A. Haar, Zur theorie der orthogonalen functionensystem, Math Annalen 69 (1910) 38; 71 (1911) 331. K. Gustafson, Partial Dierential Equations, Wiley, New York, 1987; Dover, New Jersey, 1998. K. Gustafson, B. Misra, Canonical commutation relations of quantum mechanics and stochastic regularity, Letter in Math. Phys. 1 (1976) 119. K. Gustafson, R. Goodrich, Kolmogorov systems and Haar systems, Colloq. Math. Soc. Janos Bolya 49 (1985) 401. K. Gustafson, Lectures on Computational Fluid Dynamics, Mathematical Physics, and Linear Algebra, Kaigai, Tokyo, 1996; World Scienti®c, Singapore, 1997. B. Koopman, Hamiltonian systems and transformations in Hilbert spaces, Proc. Nat. Acad. Sci. USA 17 (1931) 315. R. Goodrich, K. Gustafson, B. Misra, On converse to Koopman's Lemma, Physica A 102 (1980) 379. P. Halmos, A Hilbert Space Problem Book, Springer, New York, 1982. B.Sz. Nagy, C. Foias, Harmonic Analysis of Operators in Hilbert Space, North Holland, Amsterdam, 1970. G. Mackey, Theory of Group Representations, University of Chicago Press, Chicago, 1976. W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, in: Handbuch der Physik, S. Flugge (Eds.), Springer, Berlin, 1958. B. Misra, I. Prigogine, M. Courbage, From deterministic dynamics to probabilistic descriptions, Physica 38A (1979) 1. I. Antoniou, B. Misra, Relativistic internal Time operator, Intern. J. of Theor. Phys. 31 (1992) 119. I. Cornfeld, S. Fomin, Ya Sinai, Ergodic Theory, Springer, Berlin, 1982. C. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Springer, Berlin, 1967. P. Lax, R.S. Phillips, Scattering Theory, Academic, New York, 1967. I. Antoniou, K. Gustafson, Wavelets and Stochastic Processes, to appear. M. Stone, Linear Transformations in Hilbert Space, Amer. Math. Soc., Providence, R.I., 1932. J. Marti, Introduction to the Theory of Bases, Springer, New York, 1969. S. Ali, J.P. Antoine, J.P. Gazeau, U. Mueller, Coherent states and their generalizations: a mathematical overview, Reviews in Mathematical Physics 7 (1995) 1013. K. Flornes, A. Grossmann, M. Holschneider, B. Torresani, Wavelets on discrete ®elds, Applied Comput. Harm. Analysis 1 (1994) 137. M. Duval-Destin, M. Muschietti, B. Torresani, Continuous wavelet decompositions, multiresolutions, and contrast analysis, SIAM J. Math Anal. 24 (1993) 739. H. Stark, Continuous wavelet transform and continuous multiscale analysis, J. Math. Anal. Applic. 169 (1992) 179.
www.elsevier.nl/locate/chaos
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 Wn . 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 mechanics, 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 exposes 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 W0 of the MRA as the wandering generating subspace of the scaling transformation of the wavelet is established. Although this characterization of the wavelet subspace W0 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. *
Corresponding author. E-mail address: [email protected] (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
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In Section 3 we construct the Time operator of an arbitrary wavelet multiresolution analysis. This result is obtained by using the wandering subspace characterization obtained in Section 2 in conjunction with discrete systems of imprimitivity through which the Time operator of statistical physics is de®ned. The Time operator is seen to incorporate and exhibit in a natural way all ®ve fundamental properties of a wavelet multiresolution analysis. Then in Section 4 we address the question of extending the construction of Section 3 to a Time operator with continuous parameter. Time operators with continuous parameter occur naturally in statistical physics and quantum mechanics. Such Time operators may be viewed as playing a role analogous to that of a position operator Q in a Schr odinger couple. Therefore in Section 4 we present a new notion of continuous multiresolution analysis as naturally induced by the construction of a corresponding Time operator. In Section 5 we observe that the Haar wavelet subspaces Wn are now most naturally seen as the eigenspaces of their Time operator which by our construction is conjugate to the wavelet scaling transformations in the sense of the canonical commutation relations. From this new viewpoint the Haar basis may now be regarded most naturally as the basis of its corresponding selfadjoint Time operator, and thus as an eigenbasis whose role is to be canonically conjugate to its scaling. The same statement applies to any wavelet basis, but we highlight the Haar basis because Haar's original tasking [14,15] was to ®nd a complete orthonormal set which was not an eigenbasis derived naturally from some selfadjoint Sturm±Liouville dierential operator. Section 6 discusses our conclusions. In this paper we stay completely in L2R but extensions to higher dimensions in principle may be realized. Part of our motivation for the continuous parameter case came from an earlier work [16] in which we connected regular stationary stochastic processes to the position and momentum operators of quantum mechanical evolutions. Our motivation in the discrete parameter case came from an earlier work [17] relating Kolmogorov automorphisms and Haar Systems. We have known now for a number of years that we could obtain Time operators for wavelet structures. A preliminary announcement has been given in our lectures [18]. But it was not until we found the precise characterization given here of wavelet scaling in terms of the wandering subspace theory in the discrete parameter case, and not until we could clarify the appropriate notion of continuous wavelet multiresolution analysis given here for the continous parameter case, that we felt we had a full understanding of the essentials of this new connection of wavelets to the Time operator of statistical physics.
2. Wavelets as wandering subspsaces A discrete MRA [1±3,13] of square integrable functions over the real line consists of a hierarchy of approximations de®ned as averages on dierent scales. The ®ner the scale the better the approximation. If Pn is the orthoprojection corresponding to the approximation at a resolution 2n , the ranges Hn , n 0; 1; 2; . . . of Pn de®ne a sequence of Hilbert subspaces of L2R with the properties of MRA: (1) H T n Hn1 (2) Sn Hn f0g (3) n Hn is dense in L2R (4) A function f x is in the space Hn if and only if the scaled function f 2x is in the space Hn1 . (5) There exists a function / in H0 such that the set of translated functions /a x / x ÿ a, a 0; 1; 2; . . . is a orthonormal basis for the space H0 . We remark that this last property (5) of an MRA is stated in a number of ways in the literature, but all versions either explicitly or implicitly imply that such a function / has been chosen or can be constructed. For example it is sucient to require that one can ®nd a function g x in L2R such that the translated functions g x ÿ a form a Riesz basis of H0 . Then one can construct an orthonormal basis / x ÿ a for H0 . From the scale function / one can construct in standard ways [1±3] an oscillatory wavelet w in L2R such that the discrete translations and scalings of w provide orthonormal bases in L2R . We will not consider here the generalizations in which there are several generating functions /i , nor will we consider the multidimensional setting L2 Rn , although the new connections to dynamical systems which we establish in this paper could surely be pursued in those more general contexts.
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445
The successive multiresolution approximations of a function f are expressed through the projections Pn onto Hn : X Pn f h/na ; f i/n;a ; 1 a2Z
R 1 where hg; f i denotes the inner product ÿ1 dxg xf x. The detail or new information between two successive approximations Pn f and Pn1 f of the functions f is given by the dierence Pn1 ÿ Pn f . The range of the projection Pn1 ÿ Pn is the orthocomplement Wn of the space Hn in Hn1 Wn Hn1 Hn ;
Wn ? Hn :
2
The properties (1±5) of the MRA imply that the detail spaces Wn are also the scalings of W0 and that they are mutually orthogonal spaces generating all the space L2R . We now wish to reformulate the meaning of scaling within a wavelet multiresolution analysis. Let us recall ®rst a few basic facts. Condition (5) of the wavelet multiresolution analysis means that the function / is a cyclic vector for unitary representation of the discrete translation group Ua / x / x ÿ a, a 2 Z, on the space H0 . As the spaces Hn are just scalings of the space H0 (Condition 4), each scaled function / 2n x is also a cyclic vector for the unitary representation of the discrete translation group on the space Hn , n 2 Z. The function / gives rise therefore to the orthonormal basis /na x 2n=2 / 2n x ÿ a;
a2Z
3
of the space Hn . Next we recall the unitary representation of the ane group of translations and scalings, namely, 1 1 ÿ1 Uab f x p f bÿ1 x ÿ a p f a; b x; jbj jbj
4
where a 2 R, b 2 R ÿ f0g. Recall that the ane group R R ÿ f0g acts on the real line as follows: x7! a; bx b x a
5
i.e., ®rst translation by a 2 R and then scaling by b 6 0. The group synthesis is 0
a; b a0 ; b0 bb0 ; a0 b ÿ1 a:
6
The unit and inverse transformations are I 0; 1; a; b
ÿ1
7
ÿba; bÿ1 :
8
Since the spaces Wn are obtained from each other by scalings and are mutually orthogonal, the functions wna x 2n=2 w 2n x ÿ a;
a2Z
9
for ®xed n 2 Z form an orthonormal basis for Wn . The set wna , n 2 Z, a 2 Z is therefore an orthonormal basis for the whole space L2R . We may also express the approximation projections Pn in terms of the wavelets wna X Pn1 ÿ Pn f hwna ; f iwna : 10 a2Z
From property (4) we see that the space Hn1 is the image of the space Hn under the unitary transformation p Vf x 2f 2x: 11 The operators V n provide a unitary representation on the L2R of the discrete scaling transformations pspace S : x7!2x. We remark that the transformation Vf x 2f 2x is justpthe Koopman operator [19,20] induced by the underlying point transformation S x 2x. The factor 2 is introduced in order to have a
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unitary representation. The properties (1±3) of the multiresolution analysis are the conditions for the unitary scaling transformation V to be a bilateral shift [21]. Such unitary shifts have absolutely continuous spectrum of uniform multiplicity [22]. Thus the wavelet space W W0 H1 H0 V H0 H0
12
is exactly the wandering generating subspace of the shift operator V. The term wandering subspace from the invariant subspace theory [21,22] means that V n W ? V m W;
n 6 m
13
and the term generating means that 1
L2 R V n W: nÿ1
14
The wandering generating subspace W W0 is clearly in®nite dimensional because the functions w x ÿ a, a 2 Z are an orthonormal basis for it. As the dimension of the wandering generating subspace W is the multiplicity of the shift [21,22], we conclude that the multiresolution analysis is a bilateral shift of in®nite countable multiplicity. Therefore we have proved Theorem 1. A family of projections Pn defines a MRA of the space L2R if and only if: (1) The ranges Hn of Pn are obtained from the wandering generating subspace W of the bilateral shift V corresponding to the scaling transformation x 7! 2x through the formula: n
Hn V k W: kÿ1
15
(2) The wandering generating subspace W is a cyclic subspace for the group of translations. Each family of cyclic vectors for the translation group is a family of wavelets generating the multiresolution Pn n2Z . The multiplicity of the shift is equal to the number of linearly independent wavelets generating the multiresolution Pn n2Z . Thus a wavelet multiresolution analysis (1)±(5) is a bilateral shift of scalings on L2R with countable in®nite multiplicity such that the wandering generating subspace of the scaling shift operator is also a cyclic representation space with respect to the group of discrete translation. The cyclic vectors are the wavelets. In the next section we will employ this operator-theoretic characterization of the scaling transformation of wavelets as a part of imprimitivity relations which will de®ne a Time operator for an arbitrary wavelet MRA. Remark. Daubechies ([1], p. 129), points out that ``there are many `ladders' of spaces which satisfy the wavelet MRA properties (1)±(3),'' and that it is the additional requirement (4) that ``f 2 Vj () f 2j 2 V 0 '' that is the key requirement of multiresolution analysis. Theorem 1 may be regarded as the deeper content of that statement. In fact we believe that statement and its concrete realization in Theorem 1 deserve even further elaboration. The essence of Property (4) of a multiresolution analysis is that the subspaces Hn of Properties (1)±(3) be implemented as the exact increasing ranges of the unitary group V n . The multiplicity of H0 is transferred exactly to each Hn . The importance of the emphasis that we place upon this observation will become more apparent when we consider the continuous parameter case in Section 4. 3. The Time operator of wavelets The properties (1)±(3) of the multiresolution analysis imply that the approximation projections Pn satisfy the properties of a discrete resolution of the identity Pn , n 2 Z in the Hilbert space L2R , namely
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447
(1°) Pn < Pn1 , (2°) Pÿ1 limn!ÿ1 Pn 0, (3°) P1 limn!1 Pn I. It is also straightforward to see that property (4) of the multiresolution analysis when coupled with (1°)±(3°) means that Pn1 VPn V ÿ1 ;
16
where V is the unitary operator (11) associated to scalings and whose characterization is given in Theorem 1. The above properties (1°)±(3°), (16) are the properties of a discrete system of imprimitivity [23]. Therefore according to the de®nition of the Time operator [24±26,17] in statistical physics through such systems of imprimitivity, we are led to de®ne the Time operator of the multiresolution analysis by the formula X n Pn1 ÿ Pn : 17 T n2Z
In Eq. (17) T is a selfadjoint operator with dense domain D T . The imprimitivity conditions (1°)±(3°) (16) give the change of the Time operator at each stage n: V ÿn TV n T nI:
18
The multiresolution approximation projections Pn are the spectral projections of the Time operator T. From the wandering subspace relation (13) of the approximation projections Pn and the wavelets wna , we see that the action of the Time operator in terms of wavelets is X X n hwna ; f iwna x: 19 Tf x n2Z
a2Z
The relation of the wavelets with the Time operator is thus clear. The wavelet subspaces Wn are the age eigenspaces of the Time operator. Age n means resolution at the stage n. The analogy with statistical physics is striking because the higher the age the ®ner the detail. For any function f in its domain, the Time operator T attributes an average age which keeps step with the external time n according to hV n f ; TV n f i hf ; Tf i n:
20
Thus the following theorem has been proved. Theorem 2. Any wavelet MRA of the space L2 R defines a Time operator. The wavelet subspaces Wn of the MRA are the age eigenspaces of the Time operator. Note that the Time operator of Theorem 2 incorporates all ®ve MRA properties (1)±(5) in a natural way. Conditions (1)±(3) correspond to its spectral projections. Condition (4) corresponds to increasing time. Condition (5) by Theorem 1 couples the scaling wandering generating subspace W to the group of translations at each stage. Remark. An interesting way to view a role of the Time operator directly within the wavelet theory comes by asking the following question: given approximation projections (1°)±(3°), how may we characterize the wavelets w which are compatible with such given Pn ? We may introduce T Rn Pn1 ÿ Pn and show that any w compatible with the Pn defines a family wna of eigenfunctions of T such that Eq. (19) holds. Then any other wavelet w0 compatible with the Pn is characterized by w0na Xwna where X is unitary, XEn En X where En Pn1 ÿ Pn , and X commutes with translation. Remark. Time operators and their definition and meanings are intrinsically linked to the canonical commutation relations of quantum mechanics [24±26,16±18]. This will become more apparent in the next section where the continuous parameter wavelet case is considered. For discrete parameter as in the present section, the Time operator is defined in statistical physics [25,26] through a system of imprimitivity. In other words, the result of Theorem 2 is that wavelets determine their Time operator in exactly the same way as the Time operator occurs in statistical physics.
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4. Continuous wavelet multiresolution analysis In terms of the Time operator theory of statistical physics [16,17,24±26], it is natural to extend the results of Sections 2 and 3 to a continuous parameter. Historically, the continuous parameter wavelet theory was developed from the point of view of the wavelet transform. From that and the frame theory came discrete wavelet transforms which led to the concept of (discrete) multiresolution analysis [1±4]. Also, to our knowledge the wandering subspace theory essential to Theorems 1 and 2 has heretofore been restricted to the discrete parameter case. Thus, to our knowledge, our formulation of the concept of continuous wavelet multiresolution analysis introduced in this section for purposes of obtaining its Time operator for the continuous parameter case, is new. Our approach may be regarded in some sense as strongly reminiscent of the transition from discrete parameter Kolmogorov automorphisms to continuous parameter Kolmogorov ¯ows in the theory of stochastic processes [27]. In fact it was from this perspective [16,17] that we ®rst realized that one may obtain a Time operator for wavelet structures. We ®rst recall for the reader's convenience the following important result, which provides three formulations of the canonical commutation relations, which are fundamental to the foundation of quantum mechanics in terms of Heisenberg Algebras. The position and momentum operators in the Schr odinger representation on the space L2R have the form qf x xf x;
pf x ÿi
d x dx
21
and they satisfy the canonical commutation relation 1 p; q ÿI: i
22
Lemma 1 (CCR). Each of the following conditions is sufficient for two abstract selfadjoint operators Q; P on the Hilbert space H to be unitarily equivalent to the Schrodinger pair q; p on L2R , or to a direct sum of such pairs: 1 P ; Q ÿI i Z Q
kdPk ;
plus domain conditions;
23
Vt Pk Vt Pkt ;
24
Vt Us eÿits Us Vt ;
Vt eÿiPt ;
Vt eÿiPt ; Us eiQs :
25
The conditions (23)±(25) are often referred to as the Heisenberg, Schr odinger and Weyl, respectively, form of the canonical commutation relations. Lemma 1, often called the Stone±Von Neumann Theorem, has a long and rich history, with far-reaching implications and connections to many parts of mathematics and physics. For instance, the Weyl form may be regarded as a truncation of the Baker±Campbell± Hausdor theorem for the exponentiation of operator sums. The Schr odinger form is a version of imprimitivity systems from the theory of group representations. The Heisenberg form is the uncertainty relation from quantum mechanics. An example of the needed supplemental domain conditions in the Heisenberg form (Rellich±Dixmier): P 2 Q2 be essentially selfadjoint. For other conditions and further information, see Putnam [28]. The second result that we need to recall is key to the theory of Kolmogorov ¯ows in the ergodic theory [27] and also plays a key role in the scattering theory [29]. Lemma 2 (Von Neumann). Let there be given a strongly continuous one parameter group Vt in a Hilbert space H. Then the following conditions are equivalent.
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1. The group Vt has homogeneous Lebesgue spectrum:
449
26
2. There exists a subspace H0 of H such that i Ht Vt H0 H0 for t > 0: \ ii Vt H0 f0g:
27
ÿ1
iii
[
Vt H0 H:
ÿ1
3. There exists a strongly continuous unitary group Ut coupled to Vt through the commutation relations Vt Us eÿist Us Vt ;
ÿ1 < s; t < 1:
28
odinger couple in the sense of Lemma 2 implies that the in®nitesimal generators of Vt and Ut are a Schr Lemma 1. But Lemma 2 states more: the chain Ht is implemented by the group Vt . This will become the condition 4 of a continuous multiresolution analysis as we shall de®ne it. See our Remark at the end of Section 2 as concerns the importance of this observation. Let us now de®ne a continuous multiresolution analysis, motivated so that its properties yield a Time operator such as that obtained in Theorem 2 in the discrete parameter case. To that end we de®ne a continuous multiresolution analysis to be a continuous chain of Hilbert spaces Ht with the properties 10 H T t Hs for t < s, 20 St Ht f0g, 30 t Ht is dense in L2R , 40 Ht Vt H0 for some continuous unitary group Vt , 50 Vt is irreducible and has Lebesgue spectrum of countable multiplicity: Property (40 ) as we have stated embodies the key property of exact conjugate implementability but is otherwise permitted generality of group Vt . There are many examples of continuous MRA, notably in stochastic processes, where the increase of detail is that of r-algebra ®ltration. We have been able to show [30] in fact that all continuous multiresolution analyses may be strongly linked to such processes. It would be interesting in the future to investigate selected classes of continuous implementation groups Vt according to desired speci®c underlying wavelet group actions. In an intuitive sense [18] one such class would be a scaling action Vt f s cf bs, but technical details must be considered elsewhere. For any continuous multiresolution analysis (10 )±(50 ) we are now able to obtain a Time operator. Theorem 3. Any continuous multiresolution analysis defines a Time operator T. Proof. The (assumed right continuous) projection family Pt onto the subspaces Ht with properties (10 ), (20 ), (30 ), (50 ) is the spectral family of a selfadjoint operator with absolutely continuous Lebesgue spectrum of in®nite multiplicity. Exponentiation of that operator yields a strongly continuous unitary group Ut . By Lemma 2 the conditions (10 ), (20 ), (30 ), (40 ) guarantee the existence of a canonically conjugate continuous one parameter group Vs in the Weyl form (25,28) of the canonical commutation relations. It is straightforward to check [16,17], as in the discrete case, that the condition (40 ) means that Pkt Vt Pk Vt ; where Vt : H0 ! Ht is the unitary family of condition (40 ). Thus we are led to de®ne Z 1 T sdPs : ÿ1
29
30
By this construction, T is a ``position'' operator Q, position now being interpreted in the sense of time. For a function f in its domain D T , T attributes an average age
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hVt f ; TVt f i hf ; Tf i t
31
in which the age corresponds to the group action Vt which has been performed on f. This action or ``aging'' corresponds to the operator imprimitivity condition Vÿt TVt T tI
32
analogous to the discrete case (18). Note that the Time operator, now in the continuous parameter case, again naturally incorporates all ®ve properties 10 ± 50 of the continuous multiresolution analysis, in a meaningful way. Remark. One may obtain a more detailed representation of T in terms of cyclic vectors in the continuous parameter case, analogous to Eq. (19) in the discrete parameter case, by employing the more detailed operational calculus of unitarily equivalent selfadjoint operators with continuous spectrum [31]. See especially [31, Chs. VI, VII]. We find it especially amusing that the functions g; /; w of [31, pp. 244±250], play roles similar to those of the g; /; w of the wavelet MRA. It is of course due to the separability of L2R or H that one may obtain such spectral direct sum decompositions over the continuous spectrum. We placed an infinite multiplicity in condition (50 ) in analogy with the discrete MRA, but that need not necessarily be the case. A more detailed investigation of wavelet-like properties of such direct sum component cyclic vectors will be given elsewhere.
5. Haar's basis and the Time operator One of Haar's principal tasks in his dissertation [14] was to answer the question of whether there existed a complete orthonormal basis that was not the eigenbasis of any selfadjoint Sturm±Liouville dierential operator. This he answered armatively with his basis. This basis moreover serves as a monotone orthogonal basis for many Lp spaces [32]. With Theorem 2 we ®nd now the natural setting of Haar's basis, namely, as the eigenbasis of the selfadjoint Time operator. That is, the natural selfadjoint operator associated to the Haar basis, although not a Sturm±Liouville dierential operator, is just the Time operator of the Haar wavelets wna 2n=2 10;1 2n1 x ÿ 2a 2 ÿ 10;1 2n1 x ÿ 2a 1:
33
As we have pointed out earlier, the Time operator of any wavelet MRA may be viewed in the same way as representing the average aging by scaling of any given initial vector f. However, let us formalize this observation in terms of Haar's basis, due to its historical signi®cance as a basis constructed without regard to any given operator. Theorem 4. The natural setting of Haar's basis is that of eigenbasis of its associated Time operator. The natural role of Haar's basis is that of an eigenbasis canonically conjugate to its scaling. Remark. As Haar's basis, by his construction, becomes increasingly discontinuous with increasing n, that is the meaning of time in the Haar sense. Such a property is incompatible with needed regularity of any eigenbasis of any Sturm±Liouville boundary value problem. Remark. As we have pointed out elsewhere [15,17], Haar's three main contributions to mathematics were his generalization of Lebesgue measure to the construction of an invariant measure on any locally compact topological group, his construction of a complete orthonormal basis (33) not arriving from any regular Sturm± Liouville boundary value problem, and perhaps most overlooked, his proof that a wide class of Sturm±Liouville eigenfunction expansions converge or diverge pointwise along with the Fourier trigonometric expansion. We believe that Haar would be delighted to now see the Time operator as the correct and natural setting for his basis. Although the Time operator is not a selfadjoint second order dierential Sturm±Liouville operator, and indeed we would not want it to be such as that would violate Haar's motivation for constructing his basis,
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nonetheless the Time operator may be viewed as a ®rst order selfadjoint dierential operator, as follows. In the continuous parameter case, Section 4, by Fourier transform the Time operator may be seen to be a momentum operator p in a Schrodinger couple (21). However, as the considerations of Section 4 make clear, it is more natural to consider the Time operator as a position operator q. The same interpretation can be carried out in the discrete case, although the in®nitesimal generators are no longer p and q. The full details of this last statement must be worked out elsewhere. 6. Conclusions 1. A general connection between wavelet multiresolution analysis theory and the Time operator of statistical physics has been established. In particular, it was shown that any wavelet multiresolution analysis naturally de®nes a Time operator whose age eigenspaces are the wavelet detail subspaces Wn . 2. Intrinsic to establishing this fact is a new characterization of wavelet scaling in terms of wandering subspace theory. The generating wandering subspace is the wavelet subspace W0 . 3. Extending this Time operator construction for wavelets to the continuous parameter case led naturally to a new formulation of a continuous multiresolution analysis. In this formulation the essential property is that the wavelet chain be implemented exactly by a continuous unitary group. 4. We reinterpret Haar's basis, originally designed by him to be independent from any Sturm±Liouville dierential operator, as now naturally the eigenbasis of its corresponding wavelet Time operator. More generally, a natural interpretation of any wavelet basis is that of an eigenbasis canonically conjugate to its scaling. 5. We have learned from a recent extensive review [33] of coherent states and their generalizations that there has been concern about ``the missing link between the CWT based on the CS approach and the DWT derived from multiresolution'', [33]. An ``intermediate'' approach based upon wavelets on discrete ®elds has recently been oered [34]. We may suggest that our approach, based upon the Time operator theory of statistical physics and ergodic theory, oers a unifying view to the discrete and continuous wavelet theory. 6. Recent discussions with colleagues revealed to us two papers [35,36] which also deal, either implicitly [35] or explicitly [36], with a continuous multiresolution analysis. The ®rst paper [35] implicitly involves a notion of continuous MRA through the wavelet transform, in the earlier known sense of Calderon, Grossmann, Morlet. An interesting result [35] is that the wavelet scaling function can be expressed as a continuous in®nite product of dilated copies of a low-pass ®lter. However, it should be noted for comparison to our formulation 10 ± 50 of continuous MRA in Section IV that the work [35] does not include any such formulation. The second paper [36] is much closer to ours and does include a continuous MRA formulation: De®nition 3.1 of [36], properties (a) and (b), are essentially our 10 ± 30 . Note however that the chain Va of [36] is restricted to a the positive numbers. Properties (c) and (d) are essentially analogous to our (40 ) and (50 ), but stress instead (c) rescaling rather than (40 ) exact implementability of some continuous group Vt , and (d) translation invariance rather than (50 ) countable Lebesgue multiplicity. We originally [18] also thought in terms of the scaling group for property (40 ) but then, because our original motivation for the Time operator came from a similar construction in stochastic process theory [16], we decided that the essence of property 4 of wavelet MRAs was exact implementability by some group V n or Vt . We have commented on this point earlier in this paper. The limitations of restricting the continuous MRA to scaling groups have been investigated in [36]. In particular the generating analyzing wavelet may be nonunique and the scaling analysis approximations correspond to simple bandpass ®ltering. Thus our formulation would appear to be more general. Acknowledgements We would like to thank I. Prigogine for his interest and encouragement. This work was partly supported by the Belgian Interuniversity Attraction Poles, the NATO CR Grant 920977R and the European Commission ESPRIT Project 21042 CTIAC.
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