By A. I. Kostrikin, I. R. Shafarevich
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Additional info for Algebra IV: infinite groups, linear groups
From the definition of 1m, it follows that r l(e i(Jz)e im ·(J1T(z)dOdz. 47 ) which may be called the mth Fourier coefficient of the operator W(f). (J dO. From this formula it follows that W(fm)e bH E 82 whenever W(f)e bH E 82 . We now show that the condition W(fm)e bH E 8 2 translates into the exponential decay of the singular numbers of W(fm). To see this we calculate W(fm) by expanding 1m in terms of ~Q,,B' As 1m is -mhomogeneous and ~Q,,B is (0: - ,B)-homogeneous, we get 1m = "2)/m, ~,B,,B+m)~,B,,B+m.
Ce B11m «)!. Hence by the PaleyWiener theorem for the Fourier transform, f will be supported in Ixl ~ B. We look for a similar result in the case of the Weyl transform. Consider the multiplication operators M j and Mj , j = 1,2, ... 37 ) We calculate the Weyl transform of Mjf and Mjf in terms of W(J). We introduce the following derivations. 38 ) be the annihilation and creation operators of quantum mechanics. For a bounded operator T on L2 (lR n), we define the derivations bjT = [Aj ,T] = AjT - TAj , ~T = -[Aj,T] = TAj - AjT.
K = L Aia;of(~)a7-1 F(~) From the above we calculate k L L AiA1a;e:(~)a;Of(~)tr((a;-1 j(~))(a;-i j(~))*). 1=0 i=O But and we have the inequality As Of(~) = €ne(€~) it follows that the integral converges to 0 as € ---+ 0 unless l = 0, i = o. Therefore if we let the above, only one term survives, and we have l. tr((aj j(O))(aj j(O))*) dJi- ::; This completes the proof of the theorem. 6 The group Fourier transform An uncertainty principle on the Heisenberg group The uncertainty principle for the Fourier transform on JR n states that if a function f is concentrated, then its Fourier transform cannot be concentrated unless f is identically zero.
Algebra IV: infinite groups, linear groups by A. I. Kostrikin, I. R. Shafarevich