By Bertrand Mercier
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Additional resources for An Introduction to the Numerical Analysis of Spectral Methods
3), we have: i i~ w-Jkz. 5) where the (zj)ljl < N are the values of z. 1: u a__~t xj given by: lJl ~ N. 1: The advantage of the discrete Fourier transform is, that thanks to the existence of the Fast Fourier Transform (see, for example, Auslander-Tolimieri, the ak from the [i]), the computation of the zj can be performed in zj O(N log N) from the ak and of operations and not 46 in 0 ( N 2) full operations as one would expect when one calculates the product of a (2N+I) x (2N+I) matrix by a vector.
Periodic distribution sense is (a) The space Hr(I) P taken in the is based on the norm Ir = ( ~ llluJ ~=0 Where the Ca r of defined in section 3. L2(I) ca llu(e),l2) i/2 r ll-Jl denotes the norm are the usual binomial coefficients and We note the following result on denseness. 1: The space Proof: u s Hr(1) p ' we know that N + ~. 5). Therefore UN(e) n converges to Finally, u (a) in L2(I). r l lU-UNIll r + 0 as N + ~, a n d the result follows, since PN u e Cp(I). D. 1) shows that the operator derivative in the periodic distribution sense.
E 2N+I lJ
An Introduction to the Numerical Analysis of Spectral Methods by Bertrand Mercier