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Toeplitz matrices for the study of the fractional Laplacian on a boundedinterval. In this work we get a deep link between (--$\Delta$) $\alpha$ ]0,1[the fractional Laplacian on the interval ]0, 1[ and T N ($\Phi$ $\alpha$) theToeplitz matrices of symbol $\Phi$ $\alpha$ : $\theta$ $\rightarrow$ |1 -- ei$\theta$ | 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 functionfor the fractional equation (--$\Delta$) $\alpha$ ]0,1[ ($\psi$) = f for$\alpha$ $\in$]0, 1 2 [ and f a sufficiently smooth function on [0, 1]. Theinterest is that this Green's function is the same as the Laplacian operator oforder 2n, n $\in$ N. Mathematical Subject Classification (2000) Primary 35S05,35S10,35S11 ; Secondary 47G30.

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