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interval $-\pi:\pi$ split into M equal intervals.

midpoints are $y_K$

but i dont understand how to show

$$ \frac{1}{M}\sum_{j=1}^{M}\cos(mx_{j})=\begin{cases} 1, & \ m \equiv 0\pmod{M}\\ 0, & \text{else} \end{cases}$$

thank you very much

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1 Answer 1

up vote 2 down vote accepted

It is convenient to push the interval forward by $\pi$. Since we are working over a full period of the cosine function, the sum does not change.

Now use the fact that $$\cos x=\frac{e^{ix}+e^{-ix}}{2}.$$ Then our sum turns out to be the sum of two finite geometric series. The case $m\equiv 0\pmod{M}$ corresponds to the trivial geometric series of all $1$'s.

Some detail: After the shift, the midpoints $x'_k$ at which we evaluate $\cos(mx)$ are $x'_k=\frac{2\pi(2k+1)}{2M}$, where $k$ ranges from $0$ to $M-1$. So one of the geometric series that we calculate the sum of is $$\sum_{k=0}^{M-1}\exp\left(\frac{2\pi i(2k+1)m}{2M}\right).\tag{$1$}$$ The other is the conjugate, obtained by replacing $i$ by $-i$.

For the sum $(1)$, by taking out the obvious common factor, we can see that all we need is $\sum_{k=0}^{M-1} \exp(2\pi i m k/M)$. This reduces to finding $\sum_{k=0}^{M-1} t^k$, where $t=2\pi i m/M$. If $1-t\ne 0$, then by the usual formula for the sum of a finite geometric series, the sum is $$\frac{1-t^M}{1-t}.$$ This sum (if $1-t\ne 0$) is $0$, since $t^M=1$. The only situation in which $1-t=0$ is when $m/M$ is an integer, meaning that $m \equiv 0\pmod M$.

Remark: If we do not push forward by $\pi$, again we obtain a geometric series, with somewhat nicer symmetries. We went to the interval $[0,2\pi]$ mainly because it is more familiar.

The relevant facts can be also proved without appealing to complex numbers. What we need then is a formula for the sum of the cosines of numbers in arithmetic progression. Please see Wikipedia for the relevant formula. The formula can be proved by a telescoping series argument.

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is this the general method to whcih you said uses complex numbers?thank you again very much! also, how about if we use original interval? thank you! –  nanme Jun 18 '12 at 17:07
    
@nanme: Yes, for sines and cosines or mixtures, the complex exponential is the a very useful approach, and ultimately connects with Fourier series. I could have written the answer in terms of the original interval. In $(1)$ replace $\exp\left(\frac{2\pi i(2k+1)m}{2M}\right)$ by $\exp\left(\frac{2\pi i(2k+1)m}{2M}-\pi i\right)$. The same argument then pushes through. The transformation to $[0,2\pi]$ was made mainly for the sake of tradition! –  André Nicolas Jun 18 '12 at 18:33

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