The Fourier Transforms of the Chebyshev and Legendre Polynomials
  Analytic expressions for the Fourier transforms of the Chebyshev and Legendre polynomials are derived, and the latter is used to find a new representation for the half-order Bessel functions. The numerical implementation of the so-called unified method in the interior of a convex polygon provides an example of the applicability of these analytic expressions. 
A. S. Fokas and S. A. Smitheman
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