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This paper presents regularity results and associated high-order numericalmethods for one-dimensional Fractional-Laplacian boundary-value problems. Onthe basis of a factorization of solutions as a product of a certainedge-singular weight $\omega$ times a "regular" unknown, a characterization ofthe regularity of solutions is obtained in terms of the smoothness of thecorresponding right-hand sides. In particular, for right-hand sides which areanalytic in a Bernstein Ellipse, analyticity in the same Bernstein Ellipse isobtained for the "regular" unknown. Moreover, a sharp Sobolev regularity resultis presented which completely characterizes the co-domain of theFractional-Laplacian operator in terms of certain weighted Sobolev spacesintroduced in (Babu\v{s}ka and Guo, SIAM J. Numer. Anal. 2002). The presenttheoretical treatment relies on a full eigendecomposition for a certainweighted integral operator in terms of the Gegenbauer polynomial basis. Theproposed Gegenbauer-based Nystr\"om numerical method for theFractional-Laplacian Dirichlet problem, further, is significantly more accurateand efficient than other algorithms considered previously. The sharp errorestimates presented in this paper indicate that the proposed algorithm isspectrally accurate, with convergence rates that only depend on the smoothnessof the right-hand side. In particular, convergence is exponentially fast (resp.faster than any power of the mesh-size) for analytic (resp. in nitely smooth)right-hand sides. The properties of the algorithm are illustrated with avariety of numerical results.

Regularity theory and high order numerical methods for the (1d)-Fractional Laplacian