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Why is the following true?

$$\sum_{j=0}^{n-1}w(2\pi j/n)\left[\frac{1}{n}\sum_{k=0}^{n-1}e^{2\pi ik(j-m)/n}\right]=w(2\pi m/n)$$

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Sums are not true. Equalities may be. –  Mariano Suárez-Alvarez Mar 10 '11 at 22:39
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up vote 3 down vote accepted

$$\sum_{k=0}^{n-1}e^{2\pi ik(j-m)/n} = 0 $$ unless $j = m+ ln$ for some $l \in \mathbb{Z}$. This can be seen from a simple geometric progression argument.

Hence when you perform the outer sum only the non-zero term namely at $j=m$ remains and hence $$\sum_{j=0}^{n-1}w(2\pi j/n)\left[\frac{1}{n}\sum_{k=0}^{n-1}e^{2\pi ik(j-m)/n}\right]=w(2\pi m/n)$$

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