Given Cranck-Nicolson scheme:
$$\frac{u_{i}^{(k+1)} - u_{i}^{(k)}}{h_t}-\frac{c^2}{2}(\frac{u_{i-1}^{(k)}-2u_{i}^{(k)}+u_{i+1}^{(k)}}{h_{x}^{2}} +\frac{u_{i-1}^{(k+1)}-2u_{i}^{(k+1)}+u_{i+1}^{(k+1)}}{h_{x}^{2}}) =0$$
Is it possible to show that Cranck-Nicholson scheme is always stable with using Fourier analysis?