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By Bertrand Mercier

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Extra info for An Introduction to the Numerical Analysis of Spectral Methods

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Noting that t f d W~(s) = -W~(t-s) we have t (~N(S),U(s))ds = f ((WN(t_s),u~

2) ikl! N e ak = u(xj), Now, up to a multiplicative factor lJl ~ N. (2N+I), the (2N+I) x (2N+I) matrix of this linear system is unitary (and hence invertlble). 4) i! 1: Suppose I 2N+I Proof: Set k = wJkw-J~ = ~k£ = ~ 1 [J (N M = 2N+I is a root of order ~J = m = W k-~) • 2N+l m= of unity; then we have i if 1 0 otherwise and J if 0~ j ~ N J+M if -N ~ J < 0 j" = Since m j+M = m j we have 45 M-I 1 2N+I ~I lJ

2: coefficients, and PC u Let u e C~(1) (an)lni< N with (Uk)ke ~ as its Fourier the Fourier coefficients of its interpolant i__n_nSN; we have the relation ^ an = £12Z Un+£M where M dsf 2N+I. D. 9) to evaluate the integral defining a Fourier coefficient ^ ^ Uk, we obtain (not uk that is to say the Fourier coefficient of the interpolant of Estimation of but) a k, u. UU-PcULI0 . 14) i~u-PcUll0 < C N-riluFI for all u g H~(1). 15) u - PC u = u - PN u + Pc(PN - l)u. 16) IIPcVll 0 < CN-rllull For this purpose, we note that if the coefficients of r .

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