In a previous paper [1] it was discussed the viability of functional analysis using as a basis a couple of generic functions, and hence vectorial decomposition. Here we complete the paradigm exploiting one of the analysis methodologies developed there, but applied to phase coordinates, so needing only one function as a basis. It will be shown that, thanks to the novel iterative analysis, any function satisfying a rather loose requisite is ontologically a basis. This in turn generalizes the polar version of the Fourier theorem to an ample class of nonorthogonal bases. The main advantage of this generalization is that it inherits some of the properties of the original Fourier theorem. As a result the new transform has a wide range of applications and some remarkable consequences. The new tool will be compared with wavelets and frames. Examples of analysis and reconstruction of functions using the developed algorithms and generic bases will be given. Some of the properties, and applications that can promptly benefit from the theory, will be discussed. The implementation of a matched filter for noise suppression will be used as an example of the potential of the theory.; Comment: 11 pages

We develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation. Similar to wavelets multiresolution surfaces represent the domain boundary using a coarse control mesh and a sequence of detail vectors. Based on the multiresolution decomposition efficient and fast algorithms are available for reconstructing control meshes of varying fineness. During shape optimisation the vertex coordinates of control meshes are updated using the computed shape gradient information. By virtue of the multiresolution editing semantics, updating the coarse control mesh vertex coordinates leads to large-scale geometry changes and, conversely, updating the fine control mesh coordinates leads to small-scale geometry changes. In our computations we start by optimising the coarsest control mesh and refine it each time the cost function reaches a minimum. This approach effectively prevents the appearance of non-physical boundary geometry oscillations and control mesh pathologies, like inverted elements. Independent of the fineness of the control mesh used for optimisation, on the immersed finite element grid the domain boundary is always represented with a relatively fine control mesh of fixed resolution. With the immersed finite element method there is no need to maintain an analysis suitable domain mesh. In some of the presented two- and three-dimensional elasticity examples the topology derivative is used for creating new holes inside the domain.

We propose a class of spherical wavelet bases for the analysis of geophysical models and forthe tomographic inversion of global seismic data. Its multiresolution character allows for modeling with an effective spatial resolution that varies with position within the Earth. Our procedure is numerically efficient and can be implemented with parallel computing. We discuss two possible types of discrete wavelet transforms in the angular dimension of the cubed sphere. We discuss benefits and drawbacks of these constructions and apply them to analyze the information present in two published seismic wavespeed models of the mantle, for the statistics and power of wavelet coefficients across scales. The localization and sparsity properties of wavelet bases allow finding a sparse solution to inverse problems by iterative minimization of a combination of the $\ell_2$ norm of data fit and the $\ell_1$ norm on the wavelet coefficients. By validation with realistic synthetic experiments we illustrate the likely gains of our new approach in future inversions of finite-frequency seismic data and show its readiness for global seismic tomography.; Comment: 43 pages, 11 figures, submitted to Geophysical Journal International

A theorem of Glimm states that representation theory of an NGCR C*-algebra is always intractable, and the Cuntz algebra O_N is a case in point. The equivalence classes of irreducible representations under unitary equivalence cannot be captured with a Borel cross section. Nonetheless, we prove here that wavelet representations correspond to equivalence classes of irreducible representations of O_N, and they are effectively labeled by elements of the loop group, i.e., the group of measurable functions A:T-->U_N(C). These representations of O_N are constructed here from an orbit picture analysis of the infinite-dimensional loop group.; Comment: 6 pages, LaTeX2e "amsproc" class; expanded version of an invited lecture given by the author at the International Congress on Mathematical Physics, July 2000 in London

The area of Fourier analysis connected to signal processing theory has undergone a rapid development in the last two decades. The aspect of this development that has received the most publicity is the theory of wavelets and their relatives, which involves expansions in terms of sets of functions generated from a single function by translations and dilations. However, there has also been much progress in the related area known as \emph{time-frequency analysis} or \emph{Gabor analysis}, which involves expansions in terms of sets of functions generated from a single function by translations and modulations. In this area there are some questions of a concrete and practical nature whose study reveals connections with aspects of harmonic and functional analysis that were previously considered quite pure and perhaps rather exotic. In this expository paper, I give a survey of some of these interactions between the abstruse and the applicable. It is based on the thematic lectures which I gave at the Ninth Discussion Meeting on Harmonic Analysis at the Harish-Chandra Research Institute in Allahabad in October 2005.; Comment: 16 pages

We study the action of translation operators on wavelet subspaces. This action gives rise to an equivalence relation on the set of all wavelets. We show by explicit construction that each of the associated equivalence classes is non-empty.; Comment: 8 pages, 2 figures

In this paper we study the tail behaviour of Mexican needlets, a class of spherical wavelets introduced by Geller, Mayeli (2009). In particular, we provide an explicit upper bound depending on the resolution level j and a parameter s governing the shape of the Mexican needlets.; Comment: 25 pages

This paper provides quantitative Central Limit Theorems for nonlinear transforms of spherical random fields, in the high frequency limit. The sequences of fields that we consider are represented as smoothed averages of spherical Gaussian eigenfunctions and can be viewed as random coefficients from continuous wavelets/needlets; as such, they are of immediate interest for spherical data analysis. In particular, we focus on so-called needlets polyspectra, which are popular tools for nonGaussianity analysis in the astrophysical community, and on the area of excursion sets. Our results are based on Stein-Maliavin approximations for nonlinear transforms of Gaussian fields, and on an explicit derivation on the high-frequency limit of their variances, which may have some independent interest.

Data sets are often modeled as point clouds in $R^D$, for $D$ large. It is often assumed that the data has some interesting low-dimensional structure, for example that of a $d$-dimensional manifold $M$, with $d$ much smaller than $D$. When $M$ is simply a linear subspace, one may exploit this assumption for encoding efficiently the data by projecting onto a dictionary of $d$ vectors in $R^D$ (for example found by SVD), at a cost $(n+D)d$ for $n$ data points. When $M$ is nonlinear, there are no "explicit" constructions of dictionaries that achieve a similar efficiency: typically one uses either random dictionaries, or dictionaries obtained by black-box optimization. In this paper we construct data-dependent multi-scale dictionaries that aim at efficient encoding and manipulating of the data. Their construction is fast, and so are the algorithms that map data points to dictionary coefficients and vice versa. In addition, data points are guaranteed to have a sparse representation in terms of the dictionary. We think of dictionaries as the analogue of wavelets, but for approximating point clouds rather than functions.; Comment: Re-formatted using AMS style

Given a real, expansive dilation matrix we prove that any bandlimited function $\psi \in L^2(\mathbb{R}^n)$, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame for certain translation lattices. Moreover, there exists a dual wavelet frame generated by a finite linear combination of dilations of $\psi$ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction relies on a technical condition on $\psi$, and we exhibit a general class of function satisfying this condition.; Comment: 21 pages, 6 figures

Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces in terms of its fundamental function. In particular, we prove that these bases are greedy if and only if the Orlicz space is a Lebesgue space. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For Lebesgue spaces the approximation spaces are identified with weighted Besov spaces.; Comment: 22 pages with references

We describe how the Deaconu-Renault groupoid may be used in the study of wavelets and fractals.; Comment: To appear in "Group representations, ergodic theory, and mathematical physics: A tribute to George W. Mackey", the proceedings of the AMS special session dedicated to the memory of George W. Mackey at the January 2007 AMS meeting

We show that one can characterize the Besov spaces on a smooth compact oriented Riemannian manifold, for the full range of indices, through a knowledge of the size of frame coefficients, using the frames we have constructed in [8].; Comment: 27 pages. This paper is a continuation of our article ``Nearly Tight Frames and Space-Frequency Analysis on Compact Manifolds''. Keywords and phrases: Wavelets, Frames, Spectral Theory, Besov Spaces, Manifolds, Pseudodifferential Operators

We study a class of dynamical systems in $L^2$ spaces of infinite products $X$. Fix a compact Hausdorff space $B$. Our setting encompasses such cases when the dynamics on $X = B^\bn$ is determined by the one-sided shift in $X$, and by a given transition-operator $R$. Our results apply to any positive operator $R$ in $C(B)$ such that $R1 = 1$. From this we obtain induced measures $\Sigma$ on $X$, and we study spectral theory in the associated $L^2(X,\Sigma)$. For the second class of dynamics, we introduce a fixed endomorphism $r$ in the base space $B$, and specialize to the induced solenoid $\Sol(r)$. The solenoid $\Sol(r)$ is then naturally embedded in $X = B^\bn$, and $r$ induces an automorphism in $\Sol(r)$. The induced systems will then live in $L^2(\Sol(r), \Sigma)$. The applications include wavelet analysis, both in the classical setting of $\br^n$, and Cantor-wavelets in the setting of fractals induced by affine iterated function systems (IFS). But our solenoid analysis includes such hyperbolic systems as the Smale-Williams attractor, with the endomorphism $r$ there prescribed to preserve a foliation by meridional disks. And our setting includes the study of Julia set-attractors in complex dynamics.

We construct a wavelet and a generalised Fourier basis with respect to some fractal measures given by one-dimensional iterated function systems. In this paper we will not assume that these systems are given by linear contractions generalising in this way some previous work of Jorgensen and Dutkay to the non-linear setting.; Comment: 17 pages, 3 figures

We discuss transformation of p-adic pseudodifferential operators (in the one-dimensional and multidimensional cases) with respect to p-adic maps which correspond to automorphisms of the tree of balls in the corresponding p-adic spaces. In the dimension one we find a rule of transformation for pseudodifferential operators. In particular we find the formula of pseudodifferentiation of a composite function with respect to the Vladimirov p-adic fractional differentiation operator. We describe the frame of wavelets for the group of parabolic automorphisms of the tree of balls in the p-adic field. In many dimensions we introduce the group of mod p-affine transformations, the family of pseudodifferential operators corresponding to pseudodifferentiation along vector fields on the tree of balls in p-adic miltidimensional space and obtain a rule of transformation of the introduced pseudodifferential operators with respect to mod p-affine transformations.

We investigate the estimation of a weighted density taking the form $g=w(F)f$, where $f$ denotes an unknown density, $F$ the associated distribution function and $w$ is a known (non-negative) weight. Such a class encompasses many examples, including those arising in order statistics or when $g$ is related to the maximum or the minimum of $N$ (random or fixed) independent and identically distributed (\iid) random variables. We here construct a new adaptive non-parametric estimator for $g$ based on a plug-in approach and the wavelets methodology. For a wide class of models, we prove that it attains fast rates of convergence under the $\mathbb{L}_p$ risk with $p\ge 1$ (not only for $p = 2$ corresponding to the mean integrated squared error) over Besov balls. The theoretical findings are illustrated through several simulations.

In this paper, we introduce a generalization of the pointwise H\"older spaces. We give alternative definitions of these spaces, look at their relationship with the wavelets and introduce a notion of generalized H\"older exponent.; Comment: 18 pages

In this paper, we study nonhomogeneous wavelet systems which have close relations to the fast wavelet transform and homogeneous wavelet systems. We introduce and characterize a pair of frequency-based nonhomogeneous dual wavelet frames in the distribution space; the proposed notion enables us to completely separate the perfect reconstruction property of a wavelet system from its stability property in function spaces. The results in this paper lead to a natural explanation for the oblique extension principle, which has been widely used to construct dual wavelet frames from refinable functions, without any a priori condition on the generating wavelet functions and refinable functions. A nonhomogeneous wavelet system, which is not necessarily derived from refinable functions via a multiresolution analysis, not only has a natural multiresolution-like structure that is closely linked to the fast wavelet transform, but also plays a basic role in understanding many aspects of wavelet theory. To illustrate the flexibility and generality of the approach in this paper, we further extend our results to nonstationary wavelets with real dilation factors and to nonstationary wavelet filter banks having the perfect reconstruction property.

This paper is concerned with density estimation of directional data on the sphere. We introduce a procedure based on thresholding on a new type of spherical wavelets called {\it needlets}. We establish a minimax result and prove its optimality. We are motivated by astrophysical applications, in particular in connection with the analysis of ultra high energy cosmic rays.; Comment: 30 pages, 9 figures