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Is the sinc function both absolutely summable (L1 norm for Continuous time signals and l1 norm for Discrete time signals) and square summable (L2 norm for Continuous time signals and l2 norm for Discrete time signals) ?

Can anyone show the integration and summation calculations that calculates above norms ?

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    $\begingroup$ Is this homework? Can you give some background how this problem pops up? $\endgroup$ – Matt L. May 1 at 15:00
  • $\begingroup$ Not a homework problem. I read in a book about, and I tried to prove/disprove it, but I donot seem to be able to calculate summation and integration of sinc and its square. $\endgroup$ – nurabha May 1 at 15:11
  • $\begingroup$ it's actually hard to do directly. But, Parseval's theorem makes it trivial. So, apply that. $\endgroup$ – Marcus Müller May 1 at 18:16
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Neither the integral

$$I_1=\int_{0}^{\infty}\left|\frac{\sin(x)}{x}\right|dx\tag{1}$$

nor the sum

$$S_1=\sum_{n=1}^{\infty}\left|\frac{\sin(n)}{n}\right|\tag{2}$$

converge. So the sinc function (sequence) is not in $L_1$ ($l_1$). A proof of the fact that $I_1$ diverges can be found here. For the sum $S_1$, note that since $|\sin(n)|\ge \sin^2(n)$ (for $n\in\mathbb{R}$) we have

$$S_1\ge \sum_{n=1}^{\infty}\frac{\sin^2(n)}{n}\tag{3}$$

Here it is shown that the series on the right-hand side of $(3)$ diverges, so $S_1$ must also diverge.

The integral

$$I_2=\int_{0}^{\infty}\left(\frac{\sin(x)}{x}\right)^2dx=\frac{\pi}{2}\tag{4}$$

can be computed in many different ways. Two possibilities are shown in the answers to this question.

Also note that

$$\int_{0}^{\infty}\left(\frac{\sin(x)}{x}\right)^2dx=\int_{0}^{\infty}\frac{\sin(x)}{x}dx\tag{5}$$

The computation of the sum

$$S_2=\sum_{n=1}^{\infty}\left(\frac{\sin(n)}{n}\right)^2=\frac{\pi-1}{2}\tag{6}$$

is discussed in this question and its answers. Note that a relation similar to $(5)$ holds:

$$\sum_{n=1}^{\infty}\left(\frac{\sin(n)}{n}\right)^2=\sum_{n=1}^{\infty}\frac{\sin(n)}{n}\tag{7}$$

So clearly, the sinc function (sequence) is in $L_2$ ($l_2$).

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