Toeplitz matrices for the study of the fractional Laplacian on a bounded interval

TL;DR

This paper establishes a connection between the fractional Laplacian on a bounded interval and Toeplitz matrices, providing a Green function for certain fractional equations and linking it to higher-order Laplacians.

Contribution

It introduces a deep link between the fractional Laplacian and Toeplitz matrices, and constructs a Green function for fractional equations on an interval.

Findings

Link between fractional Laplacian and Toeplitz matrices as N approaches infinity.

Explicit Green function for fractional Laplacian on [0,1].

Green function corresponds to higher-order Laplacian operators.

Abstract

Toeplitz matrices for the study of the fractional Laplacian on a bounded interval. In this work we get a deep link between (--$\Delta$) $\alpha$ ]0,1[ the fractional Laplacian on the interval ]0, 1[ and T N ($\Phi$ $\alpha$) the Toeplitz matrices of symbol $\Phi$ $\alpha$ : $\theta$ $\rightarrow$ |1 -- e i$\theta$ | 2$\alpha$ when N goes to the infinity and for $\alpha$ $\in$]0, 1 2 [$\cup$] 1 2 , 1[. In the second part of the paper we provide a Green function for the fractional equation (--$\Delta$) $\alpha$ ]0,1[ ($\psi$) = f for $\alpha$ $\in$]0, 1 2 [ and f a sufficiently smooth function on [0, 1]. The interest is that this Green's function is the same as the Laplacian operator of order 2n, n $\in$ N. Mathematical Subject Classification (2000) Primary 35S05, 35S10,35S11 ; Secondary 47G30.