arXiv subjects

Ashley R. Zhang

Publications and source records attributed to Ashley R. Zhang.

3 recordsLinked to original sources

arXiv2025

Direct spectral problems for Paley-Wiener canonical systems

This note focuses on the direct spectral problem for canonical Hamiltonian systems on the half-line $\mathbb{R}_+$. Truncated Toeplitz operators have been effectively used to solve the inverse spectral problem when the spectral measure is a locally finite periodic measure (see \cite{MP}). Here, we reverse the inverse problem algorithm to solve the direct spectral problem for step-function Hamiltonians. For a non-step-function Hamiltonian, we consider its step-function approximations and their corresponding spectral measures, and show that these spectral measures converge to the spectral measure of the original Hamiltonian.

math.SP↗

arXiv2024

Convergence from the discrete to the continuous non-linear Fourier transform

In this note, we study the convergence from the discrete to the continuous non-linear Fourier transform. Relations between spectral problems and questions in complex function theory provide a new approach to the study of scattering problems and the non-linear Fourier transform \cite{Scatter}. In particular, the non-linear Fourier transform can be viewed from the perspective of spectral problems for differential operators. Results in \cite{MP, PZ} can be seen as results for the non-linear Fourier transform. These results are similar to some convergence problems for the discrete non-linear Fourier transform considered in \cite{T} and \cite{TT}.

math.CA↗

arXiv2020

Reduced Basis Approximations of the Solutions to Spectral Fractional Diffusion Problems

We consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction-diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction-diffusion problems. The reduced basis does not depend on the fractional power $s$ for $0<s_{\min}\leq s \leq s_{\max}<1$. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

math.NA↗