The Bessel functions *J*_{n} for even *n* look something like the sinc function. How well can you approximate the former by sums of the latter? To make things concrete, we’ll approximate *J*_{2}. Here’s a plot of *J*_{2}.

And here’s a plot of sinc(*x*) = sin(π*x*)/π*x*.

The sinc approximation for a function *f*(*x*) is given by

Sinc approximation can be remarkably accurate, nearly optimal in some sense.

The accuracy of the approximation increases as *n* gets larger and *h* gets smaller. We will fix *n* = 10. How should we pick *h*? The paper cited in this post suggests using

Let’s try that and see what happens.

The approximation isn’t very good overall, though it’s excellent near 0.

Before making plots, I had a plausible argument for why the value of *h* suggested above might be optimal. I also had an argument for why a much larger value of *h*, something on the order of 8 might be optimal. Turns out both are wrong. You can get a good approximation over a larger range by choosing *h* around 2.6.

## Related posts

- Peaks of sinc (three part series)
- Sinc and jinc sums
- Gibbs phenomenon