On the Fourier Extension of Nonperiodic Functions

Daan Huybrechs
2010 SIAM Journal on Numerical Analysis  
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 new
more » ... . We propose a new numerical method that is stable both in theory and in practice. Abstract 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 new numerical method that is stable both in theory and in practice.
doi:10.1137/090752456 fatcat:rhbkdvyhmbha3hsconm6oaej64