Here I found the proof of the DFT shift theorem and didn't get, how the summation limits can be changed:
$$\sum_{m=-\Delta}^{N-1-\Delta}x(m)e^{-j2\pi(m+\Delta)k/N}= \sum_{m=0}^{N-1}x(m)e^{-j2\pi mk/N} e^{-j2\pi k\Delta/N}$$
why it is correct? What if the values of $x(-\Delta), x(-\Delta+1), x(-\Delta+2)...$ are significantly differ from $x(0), x(1), x(2) ...$, so the total sum may output different result.