On the Fourier Extension of Nonperiodic Functions

We obtain exponentially accurate Fourier series for nonperiodic 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. Key words. Fourier series, orthogonal polynomials, least squares, frames AMS subject classifications. 42A10, 42C15, 65T40 DOI. 10.1137/090752456

• Type: Journal
• Year: 2010
• Where: SIAMNUM
• Authors: Daan Huybrechs
• Keywords: Fourier Series, Numerical Method, Orthogonal Polynomials, SIAMNUM 2010