On the Gibbs Phenomenon 1: Recovering Exponential Accuracy from the Fourier Partial Sum of a Non-Periodic Analytic Function
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING HAMPTON VA
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It is well known that the Fourier series of an analytic and periodic function, truncated after 2Nl terms, converges exponentially with N, even in the maximum norm. It is also known that if the function is not periodic, the rate of convergence deteriorates in particular there is no convergence in the maximum norm, although the function is still analytic. This is known as the Gibbs phenomenon. In this paper we show that the first 2Nl Fourier coefficients contain enough information about the function, so that an exponentially convergent approximation in the maximum norm can be constructed. Gibbs phenomenon, Fourier Series, Exponential accuracy.
- Numerical Mathematics