Related papers: Slepian Spatial-Spectral Concentration on the Ball
Spherical confinement in 3D harmonic, quartic and other higher oscillators of even order is studied. The generalized pseudospectral method is employed for accurate solution of relevant Schr\"odinger equation in an \emph{optimum,…
Obtaining constraints from the largest scales of a galaxy survey is challenging due to the survey mask allowing only partial measurement of large angular modes. This scatters information from the harmonic-space 2-point function away from…
In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order $\alpha>-1$ on the unit ball in arbitrary dimension, termed as ball PSWFs. They are eigenfunctions of both a weighted concentration integral operator,…
Spectral analysis in conjunction with discrete data in one and more dimensions can become a challenging task, because the methods are sometimes difficult to understand. This paper intends to provide an overview about the usage of the…
Current and upcoming large-scale structure surveys place stringent requirements on the mitigation of observational systematics in order to achieve their unprecedented constraining power. In this work, we investigate the potential use of the…
We consider Steklov eigenvalues of three-dimensional, nearly-spherical domains. In previous work, we have shown that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term…
Solution of some boundary value problems and initial problems in unique ball leads to the convergence and sumability problems of Fourier series of given function by eigenfunctions of Laplace operator on a sphere - spherical harmonics. Such…
This paper investigates the stability properties of the spectrum of the classical Steklov problem under domain perturbation. We find conditions which guarantee the spectral stability and we show their optimality. We emphasize the fact that…
The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer graphics and geometry processing applications. In particular, Laplacian eigenbases allow generalizing the classical Fourier analysis to manifolds. A key drawback…
We construct a pseudospectral method for the solution of time-dependent, non-linear partial differential equations on a three-dimensional spherical shell. The problem we address is the treatment of tensor fields on the sphere. As a test…
On smooth compact manifolds with smooth boundary, we first establish the sharp lower bounds for the restrictions of harmonic functions in terms of their frequency functions, by using a combination of microlocal analysis and frequency…
The spectral $k$-support norm enjoys good estimation properties in low rank matrix learning problems, empirically outperforming the trace norm. Its unit ball is the convex hull of rank $k$ matrices with unit Frobenius norm. In this paper we…
Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate…
We propose a spectral collocation method to approximate the exact boundary control of the wave equation in a square domain. The idea is to introduce a suitable approximate control problem that we solve in the finite-dimensional space of…
This paper discusses a generalization of spectral representations related to convex one-homogeneous regularization functionals, e.g. total variation or $\ell^1$-norms. Those functionals serve as a substitute for a Hilbert space structure…
We propose a transform for signals defined on the sphere that reveals their localized directional content in the spatio-spectral domain when used in conjunction with an asymmetric window function. We call this transform the directional…
Consider a family of bounded domains $\Omega_{t}$ in the plane (or more generally any Euclidean space) that depend analytically on the parameter $t$, and consider the ordinary Neumann Laplacian $\Delta_{t}$ on each of them. Then we can…
We develop a sparse spectral method for a class of fractional differential equations, posed on $\mathbb{R}$, in one dimension. These equations can include sqrt-Laplacian, Hilbert, derivative and identity terms. The numerical method utilizes…
The relations between smooth and peaked soliton solutions are reviewed for the Camassa-Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is…
We demonstrate an application of the spectral method as a numerical approximation for solving Hyperbolic PDEs. In this method a finite basis is used for approximating the solutions. In particular, we demonstrate a set of such solutions for…