A short proof of Stein's universal multiplier theorem

We give a short proof of Stein's universal multiplier theorem, purely by probabilistic methods, thus avoiding any use of harmonic analysis techniques (complex interpolation or transference methods)

Adaptive time-frequency detection and filtering for imaging in heavy clutter

Abstract. We introduce an adaptive approach for the detection of a reflector in a strongly scattering medium using a timefrequency representation of the array response matrix followed by a Singular Value Decomposition (SVD). We use the Local Cosine Transform (LCT) for the time-frequency representation and introduce a detection criterion that identifies anomalies in the top singular values, across frequencies and in different time windows, that are due to the reflector. The detection is adaptive because the time windows that contain the primary echoes from the reflector are not determined in advance. Their location and width is identified by searching through the time-frequency binary tree of the LCT. After detecting the presence of the reflector we filter the array response matrix to retain information only in the time windows that have been selected. We also project the filtered array response matrix to the subspace associated with the top singular value and then image using travel time migration. We show with extensive numerical simulations that this approach to detection and imaging works well in heavy clutter that is calibrated using random matrix theory so as to simulate regimes close to the experiments in [3]. While the detection and filtering algorithm presented here works well in general clutter it has been analyzed theoretically only for the case of randomly layered media [1]

A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem

We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrodinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for times in the NLS regime provided the initial data is suitably close to a wave packet of sufficiently small amplitude in Sobolev spaces

The mixed problem in L^p for some two-dimensional Lipschitz domains

We consider the mixed problem for the Laplace operator in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. The boundary of the domain is decomposed into two disjoint sets D and N. We suppose the Dirichlet data, f_D has one derivative in L^p(D) of the boundary and the Neumann data is in L^p(N). We find conditions on the domain and the sets D and N so that there is a p_0>1 so that for p in the interval (1,p_0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L^p

Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

We show the David-Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require its surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As a consequence every Wolff snowflake has infinite surface measure.Comment: 22 pages, 6 figure

Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II

We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second order, complex, elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning \textit{a priori} almost everywhere non-tangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying \textit{a posteriori} a separate work on bounded domains.Comment: 76 pages, new abstract and few typos corrected. The second author has changed nam

Continuous Wavelets on Compact Manifolds

Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let $\Delta_{\bf M}$ be the Laplace-Beltrami operator on ${\bf M}$. Say 0 \neq f \in \mathcal{S}(\RR^+), and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 \Delta_{\bf M})$. We show that $K_t$ is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator $f(t^2\Delta)$ on \RR^n. We define continuous ${\cal S}$-wavelets on ${\bf M}$, in such a manner that $K_t(x,y)$ satisfies this definition, because of its localization near the diagonal. Continuous ${\cal S}$-wavelets on ${\bf M}$ are analogous to continuous wavelets on \RR^n in \mathcal{S}(\RR^n). In particular, we are able to characterize the H$\ddot{o}$lder continuous functions on ${\bf M}$ by the size of their continuous ${\mathcal{S}}-$wavelet transforms, for H$\ddot{o}$lder exponents strictly between 0 and 1. If $\bf M$ is the torus \TT^2 or the sphere $S^2$, and $f(s)=se^{-s}$ (the ``Mexican hat'' situation), we obtain two explicit approximate formulas for $K_t$, one to be used when $t$ is large, and one to be used when $t$ is small

Weighted norm inequalities for polynomial expansions associated to some measures with mass points

Fourier series in orthogonal polynomials with respect to a measure $\nu$ on $[-1,1]$ are studied when $\nu$ is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in $[-1,1]$. We prove some weighted norm inequalities for the partial sum operators $S_n$, their maximal operator $S^*$ and the commutator $[M_b, S_n]$, where $M_b$ denotes the operator of pointwise multiplication by b \in \BMO. We also prove some norm inequalities for $S_n$ when $\nu$ is a sum of a Laguerre weight on $\R^+$ and a positive mass on $0$

Learning an atlas of a cognitive process in its functional geometry

Proceedings of the 22nd International Conference, IPMI 2011, Kloster Irsee, Germany, July 3-8, 2011.In this paper we construct an atlas that captures functional characteristics of a cognitive process from a population of individuals. The functional connectivity is encoded in a low-dimensional embedding space derived from a diffusion process on a graph that represents correlations of fMRI time courses. The atlas is represented by a common prior distribution for the embedded fMRI signals of all subjects. The atlas is not directly coupled to the anatomical space, and can represent functional networks that are variable in their spatial distribution. We derive an algorithm for fitting this generative model to the observed data in a population. Our results in a language fMRI study demonstrate that the method identifies coherent and functionally equivalent regions across subjects.National Science Foundation (U.S.) (IIS/CRCNS 0904625)National Science Foundation (U.S.) (CAREER grant 0642971)National Institutes of Health (U.S.) (NCRR NAC P41- RR13218)National Institute of Biomedical Imaging and Bioengineering (U.S.) (U54-EB005149)National Institutes of Health (U.S.) (U41RR019703)National Institutes of Health (U.S.) (P01CA067165)Seventh Framework Programme (European Commission) (n◦257528 (KHRESMOI)