Wavelet representations and Fock space on positive matrices

We show that every biorthogonal wavelet determines a representation by operators on Hilbert space satisfying simple identities, which captures the established relationship between orthogonal wavelets and Cuntz-algebra representations in that special case. Each of these representations is shown to have tractable finite-dimensional co-invariant doubly cyclic subspaces. Further, motivated by these representations, we introduce a general Fock-space Hilbert space construction which yields creation operators containing the Cuntz-Toeplitz isometries as a special case.     

Extensions of the Heisenberg group by one‐parameter groups of dilations which are subgroups of the affine and the symplectic groups

Extensions of the multidimensional Heisenberg group by one‐parameter groups of matrix dilations are introduced and classified up to isomorphism. Each group is isomorphic to both, a subgroup of the symplectic group and a subgroup of the affine group, and its metaplectic representation splits into two irreducible subrepresentations, each of which is equivalent to a subrepresentation of the wavelet representation.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious O (1) Gibbs oscillations in the neighborhood of edges and an overall deterioration of the unacceptable first-order convergence in rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this is achieved by mollification. Here we utilize a modified version of the two-parameter family of spectral mollifiers introduced by Gottlieb and Tadmor [GoTa85]. The ubiquitous one-parameter, finite-order mollifiers are based on dilation . In contrast, our mollifiers achieve their high resolution by an intricate process of high-order cancellation . To this end, we first implement a localization step using an edge detection procedure [GeTa00a, b]. The accurate recovery of piecewise smooth data is then carried out in the direction of smoothness away from the edges, and adaptivity is responsible for the high resolution. The resulting adaptive mollifier greatly accelerates the convergence rate, recovering piecewise analytic data within exponential accuracy while removing the spurious oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a robust, general-purpose ``black box' procedure for accurate post-processing of piecewise smooth data. March 29, 2001. Final version received: August 31, 2001.     

Symbolic Computation for Moments and Filter Coefficients of Scaling Functions

Algebraic relations between discrete and continuous moments of scaling functions are investigated based on the construction of Bell polynomials. We introduce families of scaling functions which are parametrized by moments. Filter coefficients of scaling functions and wavelets are computed with computer algebra methods (in particular Gröbner bases) using relations between moments. Moreover, we propose a novel concept for data compression based on parametrized wavelets.Received December 15, 2003     

Wavelet decompositions of L 2-functionals

Based on distribution-theoretical definitions of L ₂ and Sobolev spaces given by Werner in [P. Werner (1970). A distribution-theoretical approach to certain Lebesgue and Sobolev spaces. J. Math. Anal. Appl., 29, 19–78.] real interpolation, Besov type spaces and approximation spaces with respect to multiresolution approximations are considered. The key for the investigation are generalized moduli of smoothness introduced by Haf in [H. Haf (1992). On the approximation of functionals in Sobolev spaces by singular integrals. Applicable Analysis, 45, 295–308.]. Those moduli of smoothness allow to connect the concept of L ₂-functionals with more recent developments in multiscale analysis, see e.g. [W. Dahmen (1995). Multiscale analysis, approximation, and interpolation spaces. In: C.K. Chui and L.L. Schumaker (Eds.), Approximation Theory VIII, Vol. 2: Wavelets and Multilevel Approximation, pp. 47–88.]. In particular, we derive wavelet characterizations for the Sobolev spaces introduced by Werner and establish stable wavelet decompositions of L ₂-functionals. Generalizations to more general spaces of functionals and applications are also mentioned.     

A wavelet particle approximation for McKean–Vlasov and 2D-Navier–Stokes statistical solutions

Letting the initial condition of a PDE be random is interesting when considering complex phenomena. For 2D-Navier–Stokes equations, it is for instance an attempt to take into account the turbulence arising with high velocities and low viscosities. The solutions of these PDEs are random and their laws are called statistical solutions.     

Multiresolution schemes for conservation laws

Summary. In recent years a variety of high–order schemes for the numerical solution of conservation laws has been developed. In general, these numerical methods involve expensive flux evaluations in order to resolve discontinuities accurately. But in large parts of the flow domain the solution is smooth. Hence in these regions an unexpensive finite difference scheme suffices. In order to reduce the number of expensive flux evaluations we employ a multiresolution strategy which is similar in spirit to an approach that has been proposed by A. Harten several years ago. Concrete ingredients of this methodology have been described so far essentially for problems in a single space dimension. In order to realize such concepts for problems with several spatial dimensions and boundary fitted meshes essential deviations from previous investigations appear to be necessary though. This concerns handling the more complex interrelations of fluxes across cell interfaces, the derivation of appropriate evolution equations for multiscale representations of cell averages, stability and convergence, quantifying the compression effects by suitable adapted multiscale transformations and last but not least laying grounds for ultimately avoiding the storage of data corresponding to a full global mesh for the highest level of resolution. The objective of this paper is to develop such ingredients for any spatial dimension and block structured meshes obtained as parametric images of Cartesian grids. We conclude with some numerical results for the two–dimensional Euler equations modeling hypersonic flow around a blunt body. Received June 24, 1998 / Revised version received February 21, 2000 / Published online November 8, 2000     

Adaptive frame methods for nonlinear variational problems

In this paper we develop adaptive numerical solvers for certain nonlinear variational problems. The discretization of the variational problems is done by a suitable frame decomposition of the solution, i.e., a complete, stable, and redundant expansion. The discretization yields an equivalent nonlinear problem on the space of frame coefficients. The discrete problem is then adaptively solved using approximated nested fixed point and Richardson type iterations. We investigate the convergence, stability, and optimal complexity of the scheme. A theoretical advantage, for example, with respect to adaptive finite element schemes is that convergence and complexity results for the latter are usually hard to prove. The use of frames is further motivated by their redundancy, which, at least numerically, has been shown to improve the conditioning of the discretization matrices. Also frames are usually easier to construct than Riesz bases. We present a construction of divergence-free wavelet frames suitable for applications in fluid dynamics and magnetohydrodynamics. M. Fornasier acknowledges the financial support provided through the Intra-European Individual Marie Curie Fellowship Programme, under contract MOIF-CT-2006-039438. All of the authors acknowledge the hospitality of Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Roma “La Sapienza”, Italy, during the early preparation of this work. The authors want to thank Daniele Boffi, Dorina Mitrea, and Karsten Urban for the helpful and fruitful discussions on divergence-free function spaces.     

Anisotropic dilations of shift‐invariant subspaces and approximation properties in

Let A be an expansive linear map in . Approximation properties of shift‐invariant subspaces of when they are dilated by integer powers of A are studied. Shift‐invariant subspaces providing approximation order α or density order α associated to A are characterized. These characterizations impose certain restrictions on the behavior of the spectral function at the origin expressed in terms of the concept of point of approximate continuity. The notions of approximation order and density order associated to an isotropic dilation turn out to coincide with the classical ones introduced by de Boor, DeVore and Ron. This is no longer true when A is anisotropic. In this case the A‐dilated shift‐invariant subspaces approximate the anisotropic Sobolev space associated to A and α. Our main results are also new when S is generated by translates of a single function. The obtained results are illustrated by some examples.     

Restricted thresholding: recovering smoothness and preserving edges

Restricted non linear approximation is a generalization of the N‐term approximation in which a measure on the index set of the approximants controls the type, instead of the number, of elements in the approximation. Thresholding is a well‐known type of non linear approximation. We relate a generalized upper and lower Temlyakov property with the decreasing rate of the thresholding approximation. This relation is in the form of a characterization through some general discrete Lorentz spaces. Thus, not only we recover some results in the literature but find new ones. As an application of these results, we compress and reduce noise of some images with wavelets and shearlets and show, at least empirically, that the L²‐norm is not necessarily the best norm to measure the approximation error.     

A geometric approach to the cascade approximation operator for wavelets

This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically thecascade algorithm in wavelet theory. Let be a Hilbert space, and let be a representation ofL ( ) on . LetR be a positive operator inL ( ) such thatR(1) =1, where1 denotes the constant function 1. We study operatorsM on (bounded, but noncontractive) such that

Super‐wavelets on local fields of positive characteristic

The concept of super‐wavelet was introduced by Balan, and Han and Larson over the field of real numbers which has many applications not only in engineering branches but also in different areas of mathematics. To develop this notion on local fields having positive characteristic we obtain characterizations of super‐wavelets of finite length as well as Parseval frame multiwavelet sets of finite order in this setup. Using the group theoretical approach based on coset representatives, further we establish Shannon type multiwavelet in this perspective while providing examples of Parseval frame (multi)wavelets and (Parseval frame) super‐wavelets. In addition, we obtain necessary conditions for decomposable and extendable Parseval frame wavelets associated to Parseval frame super‐wavelets.     

A wavelet theory for local fields and related groups

Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = ℚ_p, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = ℤ_p, the ring of padic integers. Classical wavelet theories, which require a non trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of the discrete quotient Ĝ/H^⊥ to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to socalled wavelet sets in the dual group Ĝ. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.     

MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP

In this article, the properties of multiresolution analysis and self-similar tilings on the Heisenberg group are studied. Moreover, we establish a theory to construct an orthonormal Haar wavelet base in L^2（H^d） by using self-similar tilings for the acceptable dilations on the Heisenberg group.     

Radial Multiresolution, Cuntz Algebras Representations and an Application to Fractals

We study Bernoulli type convolution measures on attractor sets arising from iterated function systems on R. In particular we examine orthogonality for Hankel frequencies in the Hilbert space of square integrable functions on the attractor coming from a radial multiresolution analysis on R³. A class of fractals emerges from a finite system of contractive affine mappings on the zeros of Bessel functions. We have then fractal measures on one hand and the geometry of radial wavelets on the other hand. More generally, multiresolutions serve as an operator theoretic framework for the study of such selfsimilar structures as wavelets, fractals, and recursive basis algorithms. The purpose of the present paper is to show that this can be done for a certain Bessel–Hankel transform. Submitted: February 20, 2008., Accepted: March 6, 2008.     

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L² restrictions of Fourier transforms onto spheres in which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass in dimensions is derived.     

Commutation for Irregular Subdivision

We present a generalization of the commutation formula to irregular subdivision schemes and wavelets. We show how, in the noninterpolating case, the divided differences need to be adapted to the subdivision scheme. As an example we include the construction of an entire family of biorthogonal compactly supported irregular knot B-spline wavelets starting from Lagrangian interpolation. September 4, 1998. Date revised: July 27, 1999. Date accepted: November 16, 2000.     

Embedding theorems for classes of convex sets

Rådström's embedding theorem states that the nonempty compact convex subsets of a normed vector space can be identified with points of another normed vector space such that the embedding map is additive, positively homogeneous, and isometric. In the present paper, extensions of Rådström's embedding theorem are proven which provide additional information on the embedding space. These results include those of Hörmander who proved a similar embedding theorem for the nonempty closed bounded convex subsets of a Hausdorff locally convex vector space. In contrast to Hörmander's approach via support functionals, all embedding theorems of the present paper are proven by a refinement of Rådström's original method which is constructive and does not rely on Zorn's lemma. This paper also includes a brief discussion of some actual or potential applications of embedding theorems for classes of convex sets in probability theory, mathematical economics, interval mathematics, and related areas.