By Bertrand Mercier

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**Example text**

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

1), then I-T llu(t) -Uc(t)ll 0 < C(I+N 2) 2 llu011r, for 0 ~ t ~< T. 1), and setting WN ~ ~ - uC ~WN az ~t + LcW N = (Lc-L)u N + ~ + Lz. ), 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.

16) IIPcVll 0 < CN-rllull For this purpose, we note that if the coefficients of r . 18) lak I < ( Sn (l+n2)r/2 inequality, we have then that ~ (l+n2) -r) 1/2 ( I ( l+n2)r IVn 12) 1/2" nsY(k) n~Y(k) 51 Now, (l+n2) -r ~ CN -2r. + [ £g~/{0} 1 r b£ where 1 [k+£M )2 b£ e ~ + c---N--j. 19) with so the series ~ ~-2r ~>0 C = 2o(r). def = a(r) < +~. D. 16) follows. 3: r >i 6 . 4: when r >I 6 . We have defined the discrete Fourier transforms for only an odd number (2N+I) of points. 2) and which is of dimension x.