boundary condition impact on the Fourier stability analysis

I am looking for some reference on the stability analysis of the finite difference scheme for the linear constant coefficient pde. I have a few books and I see how the Fourier analysis is used but either periodic function is considered or the unbounded domain. Clearly, solving equation numerically, we have to impose such conditions and I am looking to see what the impact is. At least for some simple problem like $u_t=u_{xx}$ with Implicit Euler scheme and some Dirichlet boundary condition. Thanks.

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You can't typically apply Fourier analysis to non-periodic functions. So you will usually have to determine the eigenvalues of the semi-discretization. –  David Ketcheson Jan 29 at 9:24