# New PDF release: Representations of Groups with Special Consideration for the By Hermann Boerner

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24) p=l Thus, the Fourier transform of a quadratic exponential may not be a quadratic exponential, even if the number of variables is even. 4. The folIowin 9 equality is true: (o,o)k-~ (k ~! n)! Proof. -- / d~d( ((, ~)" e i(~'O)+i@'O) . 24), we have k I(O, O) = f d~d~ e~(~'~)+i(~'0}'~ i<\$'~ = H ( a + OjOj). j=l k Further, YI (a + bj) = a k + ak-l(bl + ... + bk) + ... + a~ ... bk for even elements bj. , bj~ ~. Therefore, Z(0,0) = ,=oE n! k a~ k-~ = ,=oE (k _ ~)! 25). 5. For the Laplace operator with respect to anticommutin9 variables A c" k 02 = ~ = 1 oOjooj we have the equality: a o <0,0>" = (k - ,~ + 1)~ <0,0> "-1 , n -- 1, 2 , .

To observe that both expressions for the fundamental solution coincide, one needs to make use of the formula 0~5 = (_1) m 081 ... 00rn oq~5 00rn... 12)). 22). 5. Consider the Cauchy problems for the equations ~ - = 0 - 7 and ~ - = 00 " Then OR) 5(8). However, the solutions of these problems have the same fundamental solution : g(t, 8) = ( 1 + t~-~ these problems for odd initial conditions are different (if T(0) = 0, then for the right equation, u (t, 0) = 0 + t, and for the left one, u(t, O) = 0 - t).

Observe that :F'(v)(y, ~) = f e i (y,~)+i (~,o)u (dxdO). The Fourier transform on superspace has all the properties of the usual Fourier transform (with correct consideration of right and left derivatives and right and left multiplication by a variable): 1. The direct product and the convolution of distributions are transformed into the product of the Fourier transforms. 2. 3. ~'(v)=iT(Ojv); 7[OLU~ \ ao~) = i ~ j y ' ( . ) . v~2 For even variables the usual formulas are valid . Making use of the fact that the spaces of test functions A(C~ 'rn) and E ( C ] 'm) are locally convex CSAs, we obtain that the spaces of distributions E ' ( C ] 'm) and A'(C• 'm) are locally convex CSAs with respect to convolution.

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### Representations of Groups with Special Consideration for the Needs of Modern Physics by Hermann Boerner

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