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What is the relationship between the sinc function and the Dirichlet kernel:

$$ \operatorname {sinc} (x)={\frac {\sin(x)}{x}} $$

$$ D_n(x)=\sum_{k=-n}^n e^{ikx}=1+2\sum_{k=1}^n\cos(kx)=\frac{\sin\left(\left(n +1/2\right) x \right)}{\sin(x/2)} $$

I'm primarily interested in their applications in Harmonic Analysis.

Thank you for your help!

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2 Answers

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You can think of the Dirichlet kernel as a periodic version of the sinus cardinalis: one the one hand $D_n$ is obviously periodic (with period $2\pi$). On the other hand, if $n$ is large and $x$ is small holds:

$$ \frac{\sin\left(\left(n +1/2\right) x \right)}{\sin(x/2)} \approx \frac{\sin\left(\left(n +1/2\right) x \right)}{x/2} = \frac{(n+1/2)}{2}\text{sinc}((n+1/2)x) $$

I.e. locally around $0$ it looks like a scaled sinc function. The same argument holds of course for any point $x=2\pi k$, $k\in\mathbb Z$.

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Cf. this article What is the sum of a sinc function series sampled periodically . This article excellently describes the relation between the sum of a periodic sinc sampling function and the Dirichlet kernel function.

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