I found a nice explicit approximation to the Bessel function today, using the integral:
$$J_0(x)=\frac{2}{\pi} \int_0^1 \frac{\cos x u}{\sqrt{1-u^2}}du$$
With Chebyshev-Gauss quadrature we can see that the following approximation works:
$$J_0(x) \approx \frac{1}{n} \sum_{k=1}^n \cos \left(x \cos \left( \frac{2k-1}{2n} \pi \right) \right)$$
Here's an example for $n=20$, which shows that the approximation is very good for small $x$.
However, for large arguments the quadrature quickly loses the accuracy, because of strong oscillatory behavior of the integrand.
Numerically for $ x \gg n$ the error quickly blows up and starts oscillating.
Because I want to implement this approximation in an algorithm, I would like to know if we can derive explicit bounds for the error:
$$E(n,x)=J_0(x)- \frac{1}{n} \sum_{k=1}^n \cos \left(x \cos \left( \frac{2k-1}{2n} \pi \right) \right)$$