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I want to compute the following product

$$\frac{1}{N}\sum_{t=1}^{N}\left(\sum_{s=-\infty}^{\infty}a_{s}\exp(2\pi is\frac{t}{N}\right)\left(\sum_{z=-\infty}^{\infty}a_{z}\exp(-2\pi iz\frac{t}{N})\right)$$ If $N\rightarrow\infty$ we can approximate it with integral: $$\int_{0}^{1}\left(\sum_{s=-\infty}^{\infty}a_{s}\exp(2\pi isx\right)\left(\sum_{z=-\infty}^{\infty}a_{z}\exp(2\pi izx)\right)dx=\sum_{s=-\infty}^{\infty}a_{s}\sum_{z=-\infty}^{\infty}a_{z}\int_{0}^{1}\exp(2\pi i(s-z))dx$$ Hence $s=z$ and we have $$\sum_{s=-\infty}^{\infty}a_{s}^{2}$$

On the other hand this product is equal to: $$\frac{1}{N}\left(\sum_{s=-\infty}^{\infty}a_{s}\right)\left(\sum_{z=-\infty}^{\infty}a_{z}\right)\left(\sum_{t=1}^{N}\exp(2\pi i(s-z)\frac{t}{N})\right)=\sum_{w=-\infty}^{\infty}\sum_{s=-\infty}^{\infty}a_{s}a_{s+Nw}$$

We have $$\sum_{s=-\infty}^{\infty}a_{s}^{2}+\sum_{w=-\infty,w\neq0}^{\infty}\sum_{s=-\infty}^{\infty}a_{s}a_{s+Nw}$$

Now I don't see why the second term will disappear.

Which is correct?

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  • $\begingroup$ There is an error somewhere in your penultimate formula. Could you please re-edit? Also, is this an isolated calculation or exercise? if not, could you give us a bit more background context (for instance, why you're interested)? $\endgroup$
    – Yemon Choi
    Apr 10, 2010 at 19:37
  • $\begingroup$ You are taking limits as $N\to\infty$. In the sum $\sum_{w\ne0}a_{s+Nw}$ the terms thin out as $N$ increases. Assuming $a_n\to0$ quickly enough as $n\to\pm\infty$ then $\sum_{w\ne0}a_{s+Nw}$ will tend to $0$. $\endgroup$
    – Robin Chapman
    Apr 10, 2010 at 20:02
  • $\begingroup$ Following Robin's comment, you need some hypotheses on the decay of your coefficients in order for the Fourier series to converge. $\endgroup$
    – S. Carnahan ♦
    Apr 10, 2010 at 20:11
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    $\begingroup$ Yemon Choi, thanks, I have re-edited. I have obtained this expression, while solving some least square problem. And I use Fourier series of basis functions. $\endgroup$
    – WBT
    Apr 10, 2010 at 20:17
  • $\begingroup$ Robin Chapman, Scott Carnahan In my case Fourier coefficients $a_s=sinc(s)^p$ function. But still, I am confused about the limits. Because $s->\infty$, hence $s+Nw$ can be small, depending in which order I'll take limits. $\endgroup$
    – WBT
    Apr 10, 2010 at 20:25

1 Answer 1

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This is a consequence of the Dominated Convergence Theorem, see This question.

In details: $$ \sum_{w\neq0}\sum_{s} a_s a_{s+Nw} $$ is better written $$ \sum_{s}\sum_{z} b_{s,z}^N $$ where $$ b_{s,z}^N = a_{s}a_{z} \mbox{ when } z=s \mbox{ mod } N,\, N\neq0 \mbox{ and } b_{s,z}^N=0 \mbox{ otherwise.} $$ Now, $|b_{s,z}^N|\leq |a_{s}||a_{z}|$ for all $N$, and $$\sum\sum|a_{s}||a_{z}|=(\sum |a_{s}|)^2<\infty.$$ On the other hand, for any fixed $z, s$ $$ \lim_{N\to \infty} b_{s,z}^N = 0 \,(\mbox{ since } a_{s+Nw} \to 0 \mbox{ with }N) $$ Now apply the DCT to conclude.

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