The finite Fourier transform of classical polynomials
  The finite Fourier transform of a family of orthogonal polynomials $A_{n}(x)$, is the usual transform of the polynomial extended by $0$ outside their natural domain. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families. 
Atul Dixit, Lin Jiu, Victor H. Moll and Christophe Vignat
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