arXiv subjects

Michele Berra

Publications and source records attributed to Michele Berra.

5 recordsLinked to original sources

arXiv2017

A multi-scale Gaussian beam parametrix for the wave equation: the Dirichlet boundary value problem

We present a construction of a multi-scale Gaussian beam parametrix for the Dirichlet boundary value problem associated with the wave equation, and study its convergence rate to the true solution in the highly oscillatory regime. The construction elaborates on the wave-atom parametrix of Bao, Qian, Ying, and Zhang and extends to a multi-scale setting the technique of Gaussian beam propagation from a boundary of Katchalov, Kurylev and Lassas.

math.AP↗

arXiv2016

Stockwell-Like Frames for Sobolev Spaces

We construct a family of frames describing Sobolev norm and Sobolev seminorm of the space $H^s(\mathbb{R}^d)$. Our work is inspired by the Discrete Orthonormal Stockwell Transform introduced by R.G. Stockwell, which provides a time-frequency localized version of Fourier basis of $L^2([0,1])$. This approach is a hybrid between Gabor and Wavelet frames. We construct explicit and computable examples of these frames, discussing their properties.

math.FA↗

arXiv2015

Explicit construction of non-stationary frames for $L^2$

We show the existence of a family of frames of $L^2(\mathbb{R})$ which depend on a parameter $α\in [0,1]$. If $α=0$, we recover the usual Gabor frame, if $α=1$ we obtain a frame system which is closely related to the so called DOST basis, first introduced by Stockwell and then analyzed by Battisti and Riba. If $α\in (0,1)$, the frame system is associated to a so called $α$-partitioning of the frequency domain. Restricting to the case $α=1$, we provide a truly $n$-dimensional version of the DOST basis and an associated frame of $L^2(\mathbb{R}^d)$.

math.FA↗

arXiv2015

Gabor Frames of Gaussian Beams for the Schrödinger equation

The present paper is devoted to the semiclassical analysis of linear Schrödinger equations from a Gabor frame perspective. We consider (time-dependent) smooth Hamiltonians with at most quadratic growth. Then we construct higher order parametrices for the corresponding Schrödinger equations by means of $\hbar$-Gabor frames, as recently defined by M. de Gosson, and we provide precise $L^2$-estimates of their accuracy, in terms of the Planck constant $\hbar$. Nonlinear parametrices, in the spirit of the nonlinear approximation, are also presented. Numerical experiments are exhibited to compare our results with the early literature.

math.AP↗

arXiv2014

Gabor Frame Decomposition of Evolution Operators and Applications

We compute the Gabor matrix for Schrödinger-type evolution operators. Precisely, we analyze the Heat Equation, already presented in \cite{2012arXiv1209.0945C}, giving the exact expression of the Gabor matrix which leads to better numerical evaluations. Then, using asymptotic integration techniques, we obtain an upper bound for the Gabor matrix in one-dimension for the generalized Heat Equation, new in the literature. Using Maple software, we show numeric representations of the coefficients' decay. Finally, we show the super-exponential decay of the coefficients of the Gabor matrix for the Harmonic Repulsor, together with some numerical evaluations. This work is the natural prosecution of the ideas presented in \cite{2012arXiv1209.0945C} and \cite{MR2502369}.

math.AP↗