Chebyshev domain truncation is inferior to fourier domain truncation for solving problems on an infinite interval

Domain truncation is the simple strategy of solving problems ony -, ] by using a large but finite computational interval, – L, L] Sinceu(y) is not a periodic function, spectral methods have usually employed a basis of Chebyshev polynomials,T _n(y/L). In this note, we show that becauseu(±L) must be very, very small if domain truncation is to succeed, it is always more efficient to apply a Fourier expansion instead. Roughly speaking, it requires about 100 Chebyshev polynomials to achieve the same accuracy as 64 Fourier terms. The Fourier expansion of a rapidly decaying but nonperiodic function on a large interval is also a dramatic illustration of the care that is necessary in applying asymptotic coefficient analysis. The behavior of the Fourier coefficients in the limitn for fixed intervalL isnever relevant or significant in this application.