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~ then we have that, if is continuous and periodic of period 1 u (x) - u(sx), s f(Us(X)) ~ <9,f(%)> = f l 0 so in this case we can choose v x = ~.

7) where ~ > 0 is a constant. 8) L ~ (]Rn). 4) on the derivatives of W E j we obtain ^c(~)~ k I ~ aijkW] I+--~T j, k Step 3. 10) f(w) = . 12) Re = + which is positive since we assumed that We, > 0 for all ~CA. now, use Plancherel's formula to get I~n f(wE(x))dx = ~ n f(wg(~))d~ = ~ n Re f(w~(~))d~. 5) and thus the theorem. S__tep 4. 15) aij k%j Nk I2) i=l j ,k for all % E cm and for all n c~n with In[ = I. 15) we proceed by contradiction. > n~. 18) hence J,k and therefore %~ E A + iA.

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