PRELIMINARIES AND NOTATION bounded,ic, le(u)l s Cllull for all n E H, there exists a unique ve E H so that e(u=(u, ve An operator U from H, to H2 is an isometry if (Uv, Uw)=u, w) for all U,t∈H1,U itary if moreover UHI= 02, i.e every element u2 E H2 can be written as v?= Uu, for some v1 E H,. If the en constitute an orthonormal basis in HI, and U is unitary, then the Uen constitute an orthonormal basis in H2. The reverse is also true any operator that maps an orthonormal basis to another orthonormal basis is unitary a set D is called dense in H if every u E 7 can be written as the limit of some sequence of un in D.(One then says that the closure of D is all of H. The closure of a set s is obtained by adding to it all the v that can be obtained as limits of sequences in S. )If Au is only defined for v E D, but we know that A≤Cl‖ for all 1∈D (0.09) then we can extend a to all of H "by continuity Explicitly: if u E H, find un E D so that limn-go n=1. Then the un are necessarily a Cauchy sequence, and because of (0.0.9 ), so are the Aun; the Aun have therefore a limit, which we call Au(it does not depend on the particular sequence un that was chosen) One can also deal with Inbounded operators, i.e., A for which there exists no finite C such that‖Al≤Clu‖ holds for all 1∈. It is a fact of life that these can usually only be defined on a dense set D in H, and cannot be extended with compact support, for D. The dense set on which the operator is defined is called its domain I ne adjoint A'of a bounded operator A from a Hilbert, space H, to a Hilbert space H2(which may be H1 itself)is the operator from H2 to Hi defined by (u1, Au2)=(A which should hold for all uy∈化1,u2∈2.( The existence of A'is guaranteed by Riesz'representation theorem: for fixed u2, we can define a linear functional e on Hl by e(u1)=(Au1, u2). It is clearly bounded, and corresponds therefore to a vector v so that(ul, v)=e(u). It is easy to check that the correspondence u2+v is linear; this defines the operator A. )One has A’‖=‖A|,‖AAI=‖A|2 If A'=A(only possible if A maps H to itself), then A is called self-adjoint. If a self-adjoint operator A satisfies( Au, u)>0 for all u E H, then it is' called a positive operator; this is often denoted A>0. We will write A> B if'A-B is Trace-class operators are special operators such that 2n I(Aen, en)I is finite for all orthonormal bases in H. For such a trace-class operator, >n(Aen, en)is independent of the chosen orthonormal basis; we call this sum the trace of A A=∑(Ae
PRELIMINARIES AND NOTATION If A is positive, then it is sufficient to check whether >n(Aen, en)is finite for only one orthonormal basis; if it is, then A is trace-class (This is not true for non-positive operators! The spectrum. o(A)of an operator A from H to itself consists of all the E C such that A- Ald(Id stands for the identity operator, Idu= u) does not have a bounded inverse. In a finite-dimensional Hilbert space, a(A)consists of the eigenvalues of A; in the infinite-dimensional case, a(A)contains all the eigenvalues(constituting the point spectrum)but often contains other A as well constituting the continuous spectrum.(For instance, in L(R), multiplication of f(r)with sih Tr has no point spectrum, but its continuous spectrum is -1, 11) The spectrum of a self-adjoint operator consists of only real numbers; the spec trum of a positive operator contains only non-negative numbers. The spectral radius p(a)is defined by p(A)=sup{A,A∈a(A)} It has the properties p(A)≤‖ Al and A(4= Iim All1/n 3 Self-adjoint operators can be diagonalized. This is easiest to understand if their spectrum consists only of eigenvalues(as is the case in finite dimensions) (A)={An;n∈N} with a corresponding orthonormal farmily of eigenvectors, lows that for ailu E H ∑(u.e)en=∑{ 4, Aen en=∑An,en which is the diagonalization"of A.(The spectral theorem permits us to gen- eralize this if part (or all) of the spectrum is continuous, but we will not need it in this book. )If two operators commute, i.e., ABu= B Au for all u E H, then they can be diagonalized simultaneously: there exists an orthonormal basis such en and Ben= Bne Many of these properties for bounded operators can also be formulated for un ounded operators: adjoints, spectrum, diagonalization all exist for unbounded operators as well. One has to be very careful with domains, however. For in- stance, generalizing the simultaneous diagonalization of commuting operators requires a careful definition of commuting operators: there exist pathological examples where A, B are both defined on a domain D, where ab and Ba both make sense on D and are equal on D, but where A and B nevertheless are not
XVIll PRELIMINARIES AND NOTATION simultaneously diagonalizable(because D was chosen"too small"; see, and Simon(1971)for an example). The proper definition of commutin bounded self-adjoint operators uses associated bounded operators: H comn if the ced unita adjoint operator H, the associated unitary evolution operators Ut are follows: forany vE D, the domain of H (beware: the domain of a s operator is not just any dense set on which H is well defined), UrU is th u(t)at, time t=T of the differential equation u(t)=Hu(t) with initial condition v(0)=v s Banach spaces share many properties with but are more general th generally is not derived from a scalar product ), complete with respe norm(i.e, all Cauchy sequences converge; see above). Some of the we reviewed above for Hilbert spaces also exist in Banach spaces; e.g operators, linear functionals, spectrum and spectral radius. An exa Banach space that is not a hilbert space is EP(R), the set of all functi such that‖f‖Lp(see(00.2) is finite,with1≤p<∞,P≠2. Anoth isL∞(R), the set of all bounded functions on,with‖fL∞=ssup2 The dual E. of a Banach space E is the set of all bounded linear on E; it is also a linear space, which comes with a natural norm(de (0.0.7), with respect to which it is complete: E. is a Banach space it ase of the LP-spaces, 1 Sp<oo, it turns out that elements of Lq q are related by P-I+q= l, define bounded linear functionals of one has Holder’ s inequ df(x)9(2)≤uf‖L‖lL It turns out that all bounded linear functionals on LP are of this (LP)=L9. In particular, I2 is its own dual; by Riesz,representati ( see above), every Hilbert space is its own dual. The adjoint Aof A from En to E2 is now an operator from Ef to Ei, defined by (A"2)(v1)=l2(Au) There exist different types of bases in Banach spaces. (We will hich bases are countable tute a schauder basis if, for allve e, there exist unique An E U=limN-∞∑n=1anen(ie,|u-∑n=1hen‖→0aN→∞).The requirement of the A, forces the en to be linearly independent, in th po en can be in the closure of the linear span of all the others, i. e th Tm8 o that en=limN→o msl,min )mem. In a Schauder basis, 1 of the en may be important. A basis is called unconditional if in satisfies one of the following two equivalent properties
CHAPTER The What, Why, and How of Wavelets The wavelet transforM is a tool that cuts up data or functions or operators into different frequency components, and the n studies each component with a resolu- tion matched to its seale. Forerunners of this technique were invented indlepen dently in pure inatherHatics(Calderon's resolution of the idEntity in harmonic analysis sce e g, Calderon(1904)), physics(coherent states for the (ar+ b)- group in quantuM mechanics, first constructed by Aslaksen and Klalder(1968) and linked to the hydrogen atom Hamiltonian by Paul(1985))and cngincerir (QMF filters by Esteban and Galland(1977), and later Q2MF filters with exact reconstruction property by Smith and Barnwell( 1986), Vetterli(1986)in cleG- trical engineering; wavelets were proposed for the analysis of seismic data by J. Morlet(1983)). The last five years have seen a synthesis betwee'll all these different approaches, which has been very fertile for all the fields concerned Let us stay for a moment within the signal analysis framework. (The dis- cussion can easily be translated to other fields. The wavelet transform of a signal evolving in time(e. g, the amplitude of the pressure on an tardruln, for acoustical applications )depends on two variables: scale(or frequency) and time wavelets provide a tool for tiine-frequency localization The first section tells us what time-frequency localization means and why it is of interest. The remaining sections describe different types of wavelet 1.1. Time-frequency localization In many applications, given a signal f(e)(for the moment, we assume that t is a continuous variable), one is inter in its frequency content locally in time. This is similar to music notation, for example which tells the player which notes(= frequency information)to play at any given moment. The standard ourier transform e- f(t), also gives a representation of the frequency content of f, but information con- cerning time-localization of, e.g., high frequency bursts cannot be read off easily from Ff. Time-localization can be achieved by first windowing the signal f, so
CHAPTER 1 as to cut off only a well-localized slice of and then taking its fourier transform (rw)(u,t)=/df(8)9(-t)e-w (1.11) This is the windowed Fourier transform, which is a standard technique for time- frequency localization. It is even more familiar to signal analysts in its discrete version, where t and w are assigned regularly spaced values: t= nto, W= mwo where m n range over Z, and wo, to >0 are fixed. Then(1.1.1)becomes ()=ds/(s)9(s-nto)e-imwo (1.12) This procedure is schematically represented in Figure 1. 1: for fixed n, the TwIn() correspond to the Fourier coefficients of f( )9(-nto). If, for instance g is compactly supported, then it is clear that, with appropriately chosen wo trier co oefficients Twin() are sufficient to characterize and, if need be to reconstruct f(9(-nto). Changing n amounts to shifting the"slices"by steps of to and its multiples, allowing the recovery of all of f from the Tmin(f) We will discuss this in more mathematical detail in Chapter 3. Many possible choices have been proposed for the window function g in signal analysis, most of which have compact support and reasonable smoothness. In physics,(1.1.1) is related to coherent state representations; the g (s=e'g(s-t)are the coherent states associated to the Weyl- Heisenberg group( see, e. g, Klauder and Skagerstam(1985 )) In this context, a very popular choice is a Gaussian g. In all applications, g is supposed to be well concentrated in both time and frequency; if g and g are both concentrated around zero, then (twin f (w, t)can be interpreted loosely as the"content"of f near time t and near frequency w. The windowed Fourier transform provides thus a description of f in the time-frequency plane f(tg(t) f(O) g(t-to) FIG. 1. 1 The windowed Fourer transform: the function f(t)is multiplied with the windo function g(t), and the Fourer coeffictents of the product f(t)9(t)are computed; the procedure us then repeated for translated vernons of the window, g(t-to), g(t-2to