How to construct a series convergent to the kernel of Helmholtz equation?

Question

The main problem is how to construct a symmetric sequence $$\sum_{i=1}^{N} \alpha_{i}(x)\beta_{i}(y)$$ that converges to the kernel $$\frac{e^{i\omega |x-y|}}{2i\omega}$$ in a finite interval $[a,b]$. I want to use this to do some numerical calculations. Till now I can only use the Bernstein polynomials or Fourier series, but as you will see, the convergence is not so good. So I want to ask, is there a sequence as above with better convergence? Thank you.