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We have aW N (~--~', W N) + (LcWN,W N) " ((Lc-L)UN,WN) + (~, W N) + (LZ,WN). From the antisymmetry of LC, we deduce that, (passing to the real parts) d ~ #WN' 0 ~ ~'~-~" 1 d IWNI~ = Re((Lc-L)UN,WN) + R eL-~-~rSz , W N) + Re(LZ,WN). 4) 0 and T IIWN(t)II0 < ;IWN(0)U0 + t we find Sup (I$(Lc-L)UNI]0 + ll~z~II0=~+ llLzil0). 4). ILc-L)u N = (Pc-l)a ~Du N + ~ First consider (pc_ I )au N. Let y(t) = a ~ (t); vo as a e Cp(1), we have ~N(t) Ify(t)iiT_1 < Cii~--~llT_ 1 < CilUN(t)fl~. Furthermore, from the definition of the norm li.

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. 1: liE. Ill r The space with the norm [ (i + m2) r Um Vm. II . r Proof: Suppose that u ~ Hr(1); it can be deduced that m 2~ I~m 12 < +~, 0~ e< r, mg~ Consequently u (~) g L2(1) for 0 < ~ ~ r. The converse follows immediately. D. The values definition of extended to r . Hr(1) is such that it is sensible for noninteger P The previous result permits the definition of Hr(I) to be P r ~ • • of 29 We are now in a position to give error estimates in the periodic Sobolev spaces.

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

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