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Thank you by avance for your help.

So, I found on this website, that $\sum_{n=-\infty}^{\infty} {\rm sinc}( \pi n)= 1$. But I could not find any way to prove it. I know it’s about fourrier, but I don’t know how to do so...

Does anyone know how to do prove $\sum_{n=-\infty}^{\infty} {\rm sinc}\bigl( \pi(t-n)\bigr) = 1$ ?

Thank you !

Christian Blatter
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mathenfolie
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2 Answers2

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Do you know how to prove the Fourier series of a function $f \in C^\infty[-1/2,1/2]$ converges for $x \in (-1/2,1/2)$ ?

Then look at the Fourier series of $f(x) = e^{2i \pi t x}$ at $0$.

$$c_n(f) = \int_{-1/2}^{1/2} f(x) e^{-2i \pi nx}dx = \int_{-1/2}^{1/2} e^{2i \pi (t-n)x}dx= \frac{\sin \pi (n-t)}{\pi (n-t)}$$

$$1=f(0) = \sum_n c_n(f) = \sum_n \frac{\sin \pi (n-t)}{\pi (n-t)}$$

The proof of the theorem in yellow is based on the Dirichlet kernel $$(\sum_{n=-N}^N c_n(f))-f(0) = \sum_{n=-N}^N c_n(f-f(0))=\int_{-1/2}^{1/2}(f(x)-f(0))\sum_{n=-N}^N e^{-2i \pi nx}dx$$ $$ =\int_{-1/2}^{1/2} (f(x)-f(0)) \frac{\sin(2 \pi (N+1/2) x)}{2\pi\sin(\pi x)}dx = \int_{-1/2}^{1/2} g(x) \sin(2 \pi (N+1/2) x)dx$$ $$= \int_{-1/2}^{1/2} g'(x) \frac{\cos(2 \pi (N+1/2) x)}{2 \pi (N+1/2)}dx= O(\frac1N)$$ Where $$g(x)=\frac{f(x)-f(0)}{2\pi\sin(\pi x)}=\frac{x}{2\pi\sin(\pi x)} \int_0^1 f'(xt)dt \in C^\infty[-1/2,1/2]$$

reuns
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After getting help of @reuns, here the full demonstration :

$f(x)=e^{2i\pi tx}$
$f(0)=e^0=1$

We take a period of 1.

So with $C_{n}(f)=\frac{1}{T}\int_{-T/2}^{T/2}f(t)e^{-i2\pi\frac{n}{T}t}dt$

We get : $C_{n}(f)=\int_{-1/2}^{1/2}f(t)e^{-i2n\pi t}dt$

We use $f(x)$ from above :

$= C_{n}(f)=\int_{-1/2}^{1/2}f(t)e^{-i2n\pi x}dx$

$= C_{n}(f)=\int_{-1/2}^{1/2} e^{2i\pi tx} . e^{-2i\pi nx} dx$

$= C_{n}(f)=\int_{-1/2}^{1/2} e^{2i\pi (t-n)x}dx$

$=\left [\frac{e^{2i\pi (t-n)x}}{2i\pi (t-n)} \right ]^{1/2}_{-½}$

$=\frac{e^{2i\pi (t-n)1/2} - e^{-2i\pi (t-n)1/2}}{2i\pi (t-n)}$

With Euler we know : $sin(x) = \frac{1}{2i}(e^{ix} - e^{-ix})$

So $C_n(f)= \frac{sin (\pi(t-n))}{\pi (t-n)}$

We also know that $\frac{sin(x)}{x}=sinc(x)$

Based on Dirichlet kernel : $f(x)=\sum_{n=-\infty }^{\infty}C_n(f)e^{\frac{inx2\pi}{T}}$

And $f(0)=1$

So $f(0)=\sum_{n=-\infty }^{\infty}C_n(f)e^{0}=\sum_{n=-\infty }^{\infty}\frac{sin(\pi(t-n))}{\pi(t-n)}=1$

==> $\sum_{n=-\infty }^{\infty}\frac{sin(\pi(t-n))}{\pi(t-n)}=\sum_{n=-\infty }^{\infty}sinc(\pi(t-n))=1$

mathenfolie
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