Yves Meyer

PAINLESS NONORTHOGONAL EXPANSIONS

In a Hilbert space H, discrete families of vectors {hj} with the property that f=∑j〈hj‖u2009f〉hj for every f in H are considered. This expansion formula is obviously true if the family is an orthonormal basis of H, but also can hold in situations where the hj are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non‐Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.

Some New Function Spaces and Their Applications to Harmonic Analysis

Abstract In this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications. These spaces are the “tent spaces.” They provide the natural setting for the study of such things as maximal functions (the relevant space here is T∞p), and also square functions (where the space T2p is relevant). As such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis. Thus they are closely related to Lp and Hardy spaces, important parts of whose theory become corollaries of the description of tent spaces. Also, as (“Proc. Conf. Harmonic Analysis, Cortona,” Lect. Notes in Math. Vol. 992, Springer-Verlag, Berlin/New York,1983), already indicated where these spaces first appeared explicitly, the tent spaces can be used to simplify some of the results related to the Cauchy integral on Lipschitz curves, and multilinear analysis. In retrospect one can recognize that various ideas important for tent spaces had been used, if only implicitly, for quite some time. Here one should mention Carlesons inequality, its simplifications and extensions, the theory of Hardy spaces, and atomic decompositions.

Wavelets and Applications

Some investigations conducted by (1) D. Marr in psycho-physiology of human vision, (2) J. S. Lienard in speech signal processing and (3) J. Morlet in seismic signal processing led these scientists to switch from short-time Fourier analysis to some more specific algorithms better suited to detect and analyze abrupt changes in images or signals. These algorithms are strikingly similar and in the three of them the functions e, which have a given frequency co, and are the building blocks of the standard Fourier analysis, are replaced by wavelets which are time and frequency items and are the building blocks of wavelets analysis. Wavelets have a finite duration (which can be arbitrarily small) but nevertheless, should also possess a well defined average frequency. The success of the wavelets theory is due to the remarkable formulation by A. Grossmann of J. Morlets ideas. Today this theory has applications in various branches of science whenever complicated interactions between events occuring at different scales appear. This happens in astrophysics [7] or in turbulence [3, 13, 14]. Independently of the above mentioned research, heavy constraints imposed by digital speech processing have led to the discovery of the so-called quadrature mirror filters. These filters also have some applications in image processing where they improve pyramidal algorithms [1]. During the fall of 1986, S. Mallat discovered that some quadrature mirror filters were the key to the construction of orthonormal wavelet bases generalizing the Haar system. This program was completed by I. Daubechies (1987) and A. Cohen (1990) and culminated with the discovery (1989) by G. Beylkin, R. Coifman and V. Rokhlin of striking new algorithms in numerical analysis [6]. Working on submarine passive detection, J. M. Nicolas set up a new hierarchical organization of quadrature mirror filters, distinct from the one proposed by S. Mallat. R. Coifman and the speaker proved the convergence of these schemes to new libraries of orthonormal bases resembling the waveforms used by J. S. Lienard.

Wavelets: Algorithms & Applications.

In this text, the author presents mathematical background and major wavelet applications, ranging from the digital telephone to galactic structure and creation of the universe. It discusses in detail the historic origins, the algorithms and the applications of wavelets.

Improved predictability of two-dimensional turbulent flows using wavelet packet compression

Abstract We propose to use new orthonormal wavelet packet bases, more efficient than the Fourier basis, to compress two-dimensional turbulent flows. We define the “best basis” of wavelet packets as the one which, for a given enstrophy density, condenses the L2 norm into a minimum number of non-negligible wavelet packet coefficients. Coefficients below a threshold are discarded, reducing the number of degrees of freedom. We then compare the predictability of the original flow evolution with several such reductions, varying the number of retained coefficients, either from a Fourier basis, or from the best-basis of wavelet packets. We show that for a compression ratio of 1/2, we still have a deterministic predictability using the wavelet packet best-basis, while it is lost when using the Fourier basis. Likewise, for compression ratios of 1/20 and 1/200 we still have statistical predictability using the wavelet packet best-basis, while it is lost when using the Fourier basis. In fact, the significant wavelet packet coefficients in the best-basis appear to correspond to coherent structures. The weak coefficients correspond to vorticity filaments, which are only passively advected by the coherent structures. In conclusion, the wavelet packet best-basis seems to distinguish the low-dimensional dynamically active part of the flow from the high-dimensional passive components. It gives us some hope of drastically reducing the number of degrees of freedom necessary to the computation of two-dimensional turbulent flows.

Quasicrystals, Diophantine approximation and algebraic numbers

Quasicrystals can be characterized by a remarkable Diophantine approximation property. This permits to define the dual quasicrystal Λ* as the collection of y in ℝn such that |e iy·x −1| ≤ 1 for each x in the given quasicrystal Λ. In many cases one obtains Λ** = Λ and this duality is nicely related to the spectral properties of quasicrystals.

A partial regularity result for a class of Stationary Yang-Mills Fields in high dimension

We prove, for arbitrary dimension of the base n greater than or equal to 4, stationary Yang-Mills Fields satisfying Borne approximability property are regular apart from a closed subset of the base having zero (n-4)- Hausdorff measure.

A variant of compressed sensing

This paper is motivated by some recent advances on what is now called “compressed sensing”. Let us begin with a theorem by Terence Tao. Let p be a prime number and Fp be the finite field with p elements. We denote by #E the cardinality of E ⊂ Fp. The Fourier transform of a complex valued function f defined on Fp is denoted by f . Let Mq be the collection of all f : Fp 7→ C such that the cardinality of the support of f does not exceed q. Then Terence Tao proved that for q < p/2 and for any set Ω of frequencies such that #Ω ≥ 2q, the mapping Φ : Mq 7→ l(Ω) defined by f 7→ 1Ωf is injective. Here and in what follows, 1E will denote the indicator function of the set E. This theorem is wrong if Fp is replaced by Z/NZ and if N is not a prime.

Le calcul fonctionnel sous-linéaire dans les espaces de Besov homogènes

Depuis une quinzaine d’annees, divers travaux ont ete consacres au calcul fonctionnel dans les espaces de Sobolev fractionnaires W s p (R), quand l’ordre de regularite verifie 0 < s < 1+ (1/p). Ces espaces partagent la propriete d’admettre un calcul fonctionnel sous-lineaire, au sens ou il existe des fonctions non affines f : R → R telles que l’on ait, pour une certaine constante c et toute fonction g ∈ W s p , ‖f ◦ g‖ ≤ c‖g‖ , (1)