# Bertrand Mercier's An Introduction to the Numerical Analysis of Spectral PDF By Bertrand Mercier

ISBN-10: 3540511067

ISBN-13: 9783540511069

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Additional resources for An Introduction to the Numerical Analysis of Spectral Methods

Example text

E 2N+I lJ

5)) with W = e We have N 2i~/2N PC : C0(1) + SN' zj = u(~j). 7)) N (u,v)N = 2-~ 1 Finally, ~ j=-N+I as the numerical u(~j) v(~j) . 8) (the proof is left to the reader as an exercise). 1, namely llu-PCutl 0 < C(I+N2) -r/2 llullr. 1: II vN Us Proof: l,VNl,2 = For a ~ s, we have the "inverse" inequality (I+N2)(s-~)/211VNH We have. for for all v N e S N. vN e SN: [ (l+m2)Slvm 12 < (I+N2) s-O ~ (l+m2)°IVm 62 = (l+N2)S-O,lVN,t2 ImI

26 following Proposition 2. 6), where Wn,8(w ) dsf Wn(e-w) = e in0 e -inw. Finally in@ (f*Wn)(@) = e ^ = fn Wn(8)' and so ^ f = lim ~ N+~ I n ~ N f W . D. 6. Perlodie Sobolev Spaces Let I following be the fashion; for interval ]-~,~[. We u s L2(1), we set ilulir = ( I (l+m2)rl~mi2) I~ me ^ where u are the Fourier coefficients m We define the space define of u. ,r}, Hr(1) P where the derivative denoted by the superscript see section 5). 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.