What is known about sums of the form $\sum_{n=-\infty}^{\infty} \operatorname{sinc} (n^{p})$?

Solution 1:

Not a complete answer, but it' too much content for a comment, and hopefully this can help someone else find the answer! This a proof for the integrals. I don't know what to do for the sums.

For the integrals:

$$\int_{\mathbb R}\text{sinc}(x^p)dx = 2 \int_0^{+\infty}\text{sinc}(x^p)dx=\frac{2}{p}\int_0^{+\infty}\text{sinc}(y)y^{\frac 1 p -1}dy=\frac 1 p \int_{\mathbb R}\text{sinc}(y)|y|^{\frac 1 p -1}dy$$ The Fourier transform of $y\mapsto |y|^\alpha$ is $-2\frac{\sin\left(\frac {\pi \alpha}{2}\right)\Gamma(\alpha +1)}{|2\pi \xi|^{\alpha +1}}$

With the definition that $\text{sinc}(x)=\frac {\sin x}x$, the Fourier transform of $\text{sinc}$ is $\xi\mapsto \pi 1_{|\xi|<\frac 1 {2\pi}}(\xi)$

Thus using the Plancherel theorem: $$\int_{\mathbb R}\text{sinc}(x^p)dx=-\frac 2 p\int_{-\frac 1 {2\pi}}^{\frac 1 {2\pi}}\frac{\sin\left(\frac {\pi \left(\frac 1 p -1 \right)}{2}\right)\Gamma(\frac 1 p)}{|2\pi \xi|^{\frac 1 p}}\pi d\xi=\frac{2\cos\left(\frac {\pi}{2p}\right)\Gamma\left(\frac 1 p\right)}{p-1}$$

For the sums:

I honestly don't know how and if they can be computed. My instinct was again to use Fourier theory and write $$ \sum_{n\in\mathbb Z}g(n)\frac{\sin(n^p)}{n^p} =\pi\sum_{n\in\mathbb Z}g(n)\int_{-\frac 1 {2\pi}}^{\frac 1 {2\pi}}e^{2i\pi\xi n^p}\,d\xi =\pi\int_{-\frac 1 {2\pi}}^{\frac 1 {2\pi}}\sum_{n\in\mathbb Z}g(n)e^{2i\pi\xi n^p}\,d\xi\tag{1}$$

So I was wondering if the (lacunary?) Fourier series $$\sum_{n\in\mathbb Z}e^{2i\pi\xi n^p}$$ could be computed (or at least averaged) in some ways. I assume this doesn't have closed form though (maybe related to the Jacobi theta function?).

Sorry I'm out of ideas here.