Volume 39, Issue 8, August 1998
 SPECIAL ISSUE: WAVELET AND TIMEFREQUENCY ANALYSIS


Introduction to the special issue on wavelet and timefrequency analysis
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A general theorem on squareintegrability: Vector coherent states
View Description Hide DescriptionWe derive a generalization of the wellknown theorem for the square integrability of a unitary irreducible representation of a locally compact group. The generalization covers the case of representations admitting vector coherent states. The result is illustrated by an example drawn from the isochronous Galilei group. The construction yields a wide variety of coherent states, labeled by phase space points, which satisfy a resolution of the identity condition, and incorporate spin degrees of freedom.

Wavelet transforms and discrete frames associated to semidirect products
View Description Hide DescriptionWe consider a semidirect product and its unitary representations of the form where Ind is the unitary induction, is in the dual group of is the stability group of and is a unitary representation of We give sufficient conditions such that defines a wavelet transform and a discrete frame.

Continuous wavelet transforms with Abelian dilation groups
View Description Hide DescriptionThis paper is devoted to the construction of wavelet (or coherent state) systems arising from the action of certain semidirect products on For this purpose previous results which guarantee the existence of inversion formulas are applied to the special case where is Abelian. The questions of systematic construction and conjugacy of such groups are completely resolved by setting up a correspondence to unit groups of commutative associative algebras. As an application the numbers of conjugacy classes of possible Abelian groups are computed for For there are uncountably many conjugacy classes. We then compute the admissibility conditions belonging to Abelian groups. The final section contains a characterization of Abelian matrix groups acting ergodically on some subset of This result ensures that the approach via associative algebras yields all possible groups.

Wavelets on the sphere and related manifolds
View Description Hide DescriptionWe present a purely grouptheoretical derivation of the continuous wavelet transform (CWT) on the sphere based on the construction of general coherent states associated to square integrable group representations. The parameter space of the CWT, is embedded into the generalized Lorentz group via the Iwasawa decomposition, so that where Then the CWT on is derived from a suitable unitary representation of acting in the space of finite energy signals on which turns out to be square integrable over We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition, which entails all the usual filtering properties of the CWT. Next the Euclidean limit of this CWT on is obtained by redoing the construction on a sphere of radius and performing a group contraction for from which one recovers the usual CWT on flat Euclidean space. Finally, we discuss the extension of this construction to the twosheeted hyperboloid and some other Riemannian symmetric spaces.

Weighted trace formula near a hyperbolic trajectory and complex orbits
View Description Hide DescriptionIn this paper we consider a weighted trace formula for Schrödinger operators. More precisely, let and denote the eigenfunctions and eigenvalues of a Schrödingertype operator with a discrete spectrum. Let be a coherent state centered at a point of a hyperbolic closed orbit γ. We show that, as ℏ→0, the leading term of can be expressed in terms of the analytic continuation on the upper and lower halfplanes of the positive and negative frequencies part of φ. The result is also related to complex trajectories surrounding γ.

Separability, positivity, and minimum uncertainty in time–frequency energy distributions
View Description Hide DescriptionGaussian signals play a very special role in classical time–frequency analysis because they are solutions of apparently unrelated problems such as minimum uncertainty, and positivity and separability of Wigner–Ville distributions. We investigate here some of the logical connections which exist between these different features, and we discuss some examples and counterexamples of their extension to more general joint distributions within Cohen’s class and the affine class.

Timefrequency transfer function calculus (symbolic calculus) of linear timevarying systems (linear operators) based on a generalized underspread theory
View Description Hide DescriptionWe introduce a generalized concept of underspread linear timevarying systems (linear operators) which contains a previous definition as a special case. We show that an existing approximate transfer function calculus (symbolic calculus) can be extended to this wider and practically more relevant class of underspread operators. As a mathematical underpinning of this calculus, we establish explicit bounds on various error quantities associated with it. The transfer function calculus provides a theoretical basis for various methods recently proposed for nonstationary signal processing, and it has important implications in the theory of timevarying power spectra.

Symbolic calculus on the timefrequency halfplane
View Description Hide DescriptionThis study concerns a special symbolic calculus of interest for signal analysis. This calculus associates functions on the timefrequency halfplane with linear operators defined on the positivefrequency signals. Full attention is given to its construction which is entirely based on the study of the affine group in a simple and direct way. The correspondence rule is detailed and the associated Wigner function is given. Formulas expressing the basic operation (starbracket) of the Lie algebra of symbols, which is isomorphic to that of the operators, are obtained. In addition, it is shown that the resulting calculus is covariant under a threeparameter group which contains the affine group as subgroup. This observation is the starting point of an investigation leading to a whole class of symbolic calculi which can be considered as modifications of the original one.

Wavelet analysis of signals with gaps
View Description Hide DescriptionA recently introduced algorithm [Frick et al., Astrophys. J. 483, 426 (1997)] of spectral analysis of data with gaps via a modified continuous wavelet transform is developed and studied. This algorithm is based on a family of functions called "gapped wavelets" which fulfill the admissibility condition on the gapped support. The wavelet family is characterized by an additional parameter which should be calculated for every scale and position. Three theorems concerning the properties of gapped wavelet transform are formulated and proved. They affirm the global stability of the algorithm as well as its stability in both limits of large and small scales. These results are illustrated by some numerical examples, which show that the algorithm really attenuates the artifacts coming from gaps (and/or boundaries), and is particularly efficient at small and large scales.

Some remarks on the Navier–Stokes equations in
View Description Hide DescriptionWe study various existence and uniqueness results for solutions of the Navier–Stokes equations in connection with function spaces related to real harmonic analysis.

Formal improvements in the solution of the wavelettransformed Poisson and diffusion equations
View Description Hide DescriptionHermitian wavelets’ relation to the Laplace operator leads to a natural measure of the scale factor that emphasizes the largest component wave number. For the Poisson equation (e.g. the pressure equation in Navier–Stokes turbulence), the wavelet transform of the solution at a given location and scale depends on the wavelet transform of the source field at the same location and at nearby and larger scales. For the diffusion problem, the Hamiltonian formulation is simplified through a canonical transformation.

Oscillation spaces: Properties and applications to fractal and multifractal functions
View Description Hide DescriptionWe establish a wavelet characterization of the oscillation, and give two applications: a wavelet formula for the fractal dimension of graphs and a multifractal formalism for “chirptype” Hölder singularities.

Random cascades on wavelet dyadic trees
View Description Hide DescriptionWe introduce a new class of random fractal functions using the orthogonal wavelet transform. These functions are built recursively in the spacescale halfplane of the orthogonal wavelet transform, “cascading” from an arbitrary given large scale towards small scales. To each random fractal function corresponds a random cascading process (referred to as a cascade) on the dyadic tree of its orthogonal wavelet coefficients. We discuss the convergence of these cascades and the regularity of the soobtained random functions by studying the support of their singularity spectra. Then, we show that very different statistical quantities such as correlation functions on the wavelet coefficients or the waveletbased multifractal formalism partition functions can be used to characterize very precisely the underlying cascading process. We illustrate all our results on various numerical examples.

Timedependent scattering on fractal measures
View Description Hide DescriptionIn this paper we study the time evolution for the Schrödinger equation and the wave equation on the line when the interaction term is a fractalmeasure. First, we extend the usual onedimensional potential scattering formalism to interactions defined as measures. Then we show how to retrieve information on the fractality of the interaction term from timedependent scattering data. In the case of the Schrödinger equation we shall obtain the wavelet correlation dimension of the scatterer. For the wave equation the whole set of generalized multifractal dimensions can be recovered, provided the scatterer actually is fractal (nonsmooth). In this latter case, we also show how the reflected wave packets can be interpreted in terms of wavelet transform of the interaction.

Orthonormal sets of localized functions for a Landau level
View Description Hide DescriptionMaximal sets of commuting magnetic translations are used for constructing a set of eigenfunctions for a Landau level on a von Neumann–Gabor lattice. Localization and orthogonality turn out to be two conflicting features of this set. It is shown how to construct complete orthonormal sets of optimally localized eigenfunctions on a von Neumann–Gabor lattice for each Landau level. By using the Balian–Low theorem it is pointed out that the uncertainties of the orbit center coordinates cannot both be made finite.

Tauwavelets in the plane
View Description Hide DescriptionFirst an iterative process is described for subdivision and for scalingup of the Penrose–Robinson tiles. Then we build the τwavelets of Haar (i.e., suitably normalized characteristic functions) on the tiles so that the orthogonality and multiresolution conditions are satisfied. Here

Wavelets bases adapted to a selfsimilar quasicrystal
View Description Hide DescriptionGiven any selfsimilar quasicrystal Λ in with inflation we construct bases of having the following structure: where the mother wavelets are smooth and with exponential decay or compact support. We also show that wavelets constitute a relatively compact set in some Sobolev space and that they depend continuously on λ when Λ is equipped with an appropriate topology.

Square wave analysis
View Description Hide DescriptionThis paper concerns square wave analysis, which is in no small part a generalization of the Fourier analysis based on sine and cosine functions. Following sine and cosine functions square waves become another frequently used and easily generated waveform in electronics, so it is an urgent practical problem to study the basic properties of square waves and the fundamental theory of square wave analysis. Like the sinecosine function system, the square wave system, i.e., the function system of square waves with different frequencies (or periods), is linearly independent and complete in the real Hilbert spaces, so square waves can approximate an arbitrary quadratically integrable function with a vanishing meansquare error. Since the square wave system is nonorthogonal, its biorthogonal functions are presented so that a function with a certain condition can be expanded as a square wave series easily. Meanwhile the orthogonalization of the square wave system is discussed so that a quadratically integrable function can be approximated best by a superposition of finite square waves. These results form the theory basis of the square wave analysis technique in modern electronics.
