Supercritical Mean Field Equations on convex domains and the Onsager's statistical description of two-dimensional turbulence

We are motivated by the study of the Microcanonical Variational Principle within the Onsager's description of two-dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails. We obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding Mean Field Equation on convex and "thin" enough domains in the supercritical (with respect to the Moser-Trudinger inequality) regime. This is a brand new achievement since existence results in the supercritical region were previously known \un{only} on multiply connected domains. Then we study the structure of these solutions by the analysis of their linearized problems and also obtain a new uniqueness result for solutions of the Mean Field Equation on thin domains whose energy is uniformly bounded from above. Finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and use it together with all the results obtained so far to solve the Microcanonical Variational Principle in a small range of supercritical energies where the entropy is eventually shown to be concave.Comment: 35 pages. In this version we have added an interesting remark (please see Remark 1.17 p. 9). We have also slightly modified the statement of Proposition 1.14 at p.8 so to include a part of it in a separate 4-line Remark just after it (please see Remark 1.15 p.9

Radon transforms: Unitarization, Inversion and Wavefront sets

The first contribution of this thesis is a new approach based on the theory of group representations in order to solve in a general an unified way the unitarization and inversion problems for generalized Radon transform associated to dual pairs (G/K,G/H) of homogeneous spaces of a locally compact group G, where K and H are closed subgroups of G. Precisely, under some technical assumptions, if the quasi-regular representations of G acting on L^2(G/K) and L^2(G/H) are irreducible, then the Radon transform, up to a composition with a suitable pseudo-differential operator, can be extended to a unitary operator intertwining the two representations. If, in addition, the representations are square integrable, an inversion formula for the Radon transform based on the voice transform associated to these representations is given. Several examples are discussed. The second purpose of the thesis is to investigate the connection between the shearlet transform and the wavelet transform, which has to be found in the Radon transform in affine coordinates. This link yields a formula for the shearlet transform that involves only integral transforms applied to the affine Radon transform of the signal, thereby opening new perspectives both for finding a new algorithm to compute the shearlet transform of a signal and for the inversion of the Radon transform. Furthermore, we study the role of the Radon transform in microlocal analysis, especially in the resolution of the wavefront set in shearlet analysis. We propose a new approach based on the wavelet transform and on the Radon transform which clarifies how the ability of the shearlet transform to characterize the wavefront set of signals follows directly by the combination of the microlocal properties inhereted by the one-dimensional wavelet transform with a sensitivity for directions inhereted by the Radon transform. Finally, the last chapter of the thesis is devoted to the extension of the shearlet transform to distributions. Our main results are continuity theorems for the shearlet transform and its transpose, called the shearlet synthesis operator, on various test function spaces. Then, we use these continuity results to develop a distributional framework for the shearlet transform via a duality approach. This work arises from the lack in the theory of a complete distributional framework for the shearlet transform and from the link between the shearlet transform with the Radon and the wavelet transforms, whose distribution theory is a deeply investigated and well known subject in applied mathematics

Composite likelihood inference in a discrete latent variable model for two-way "clustering-by-segmentation" problems

We consider a discrete latent variable model for two-way data arrays, which allows one to simultaneously produce clusters along one of the data dimensions (e.g. exchangeable observational units or features) and contiguous groups, or segments, along the other (e.g. consecutively ordered times or locations). The model relies on a hidden Markov structure but, given its complexity, cannot be estimated by full maximum likelihood. We therefore introduce composite likelihood methodology based on considering different subsets of the data. The proposed approach is illustrated by simulation, and with an application to genomic data

Phase retrieval of bandlimited functions for the wavelet transform

We study the recovery of square-integrable signals from the absolute values of their wavelet transforms, also called wavelet phase retrieval. We present a new uniqueness result for wavelet phase retrieval. To be precise, we show that any wavelet with finitely many vanishing moments allows for the unique recovery of real-valued bandlimited signals up to global sign. Additionally, we present the first uniqueness result for sampled wavelet phase retrieval in which the underlying wavelets are allowed to be complex-valued and we present a uniqueness result for phase retrieval from sampled Cauchy wavelet transform measurements.Comment: 18 page

Unique wavelet sign retrieval from samples without bandlimiting

We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multi-wavelet frame coefficients $$\{\lvert \mathcal{W}_{\phi_i} f(\alpha^{m}\beta n,\alpha^{m}) \rvert: i\in\{1,2,3\}, m,n\in\mathbb{Z}\}$$ for every $\alpha>1,\beta>0$ with $\beta\ln(\alpha)\leq 4\pi/(1+4p)$, $p>0$, when the three wavelets $\phi_i$ are suitable linear combinations of the Poisson wavelet $P_p$ of order $p$ and its Hilbert transform $\mathscr{H}P_p$. For complex-valued signals we find that this is not possible for any choice of the parameters $\alpha>1,\beta>0$ and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients.Comment: 13 pages, 2 figure

Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in L2(R)L^2(\mathbb{R})

Sampled Gabor phase retrieval - the problem of recovering a square-integrable signal from the magnitude of its Gabor transform sampled on a lattice - is a fundamental problem in signal processing, with important applications in areas such as imaging and audio processing. Recently, a classification of square-integrable signals which are not phase retrievable from Gabor measurements on parallel lines has been presented. This classification was used to exhibit a family of counterexamples to uniqueness in sampled Gabor phase retrieval. Here, we show that the set of counterexamples to uniqueness in sampled Gabor phase retrieval is dense in $L^2(\mathbb{R})$, but is not equal to the whole of $L^2(\mathbb{R})$ in general. Overall, our work contributes to a better understanding of the fundamental limits of sampled Gabor phase retrieval.Comment: 5 pages, 2 figure

Unitarization of the Horocyclic Radon Transform on Homogeneous Trees

Following previous work in the continuous setup, we construct the unitarization of the horocyclic Radon transform on a homogeneous tree X and we show that it intertwines the quasi regular representations of the group of isometries of X on the tree itself and on the space of horocycles.Comment: 16 pages, 3 figure