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We obtain exponentially accurate Fourier series for non-periodic functions on the interval [−1, 1] by extending these functions to periodic functions on a larger domain. The series may be evaluated, but not constructed, by means of the FFT. A complete convergence theory is given based on orthogonal polynomials that resemble Chebyshev polynomials of the first and second kinds. We analyze a previously proposed numerical method, which is unstable in theory but stable in practice. We propose a newdoi:10.1137/090752456 fatcat:rhbkdvyhmbha3hsconm6oaej64