Just wondered what this might be and the result is pretty interesting at least in the discrete case. Here it goes:
Consider a funciton defined on . The discrete Fourier transform is given by:
The discrete Fourier transform of the discrete Fourier transform is given by
Now, the sum is special. Doing the usual geometric progression thing:
So, if is not a multiple of or then for that pair of . When , only for , and this value can be easily computed to be . Therefore, When , only makes and only this term would remain in the sum. A simple computation yields . Simlarly it can be shown that . Putting it all together:
What is interesting is that to compute the values of the original function one can also do a forward transform (instead of the inverse transform).
Upon further thought I found that there is a much simpler one-line proof:
the case of being trivial. So, ofcourse the forward and backward transforms of the fourier transform are related.