Spectral methods work (simplified) as follows. Consider the problem \begin{align} \partial_t u(t,x) = \mathcal{L} u(t,x) \end{align} where $\mathcal{L}$ is some differential operator. We then try to express $u$ in some (polynomial) basis, where the basis functions are eigenfunctions of the differential operator. We express $u$ as \begin{align} u = \sum_{i=0}^\infty \hat u_i(t) \phi_i(x) \end{align} and the approximation \begin{align} u^N = \sum_{i=0}^N \hat u_i(t) \phi_i(x) \end{align} where \begin{align} \mathcal{L} \phi_i = \lambda_i \phi_i \end{align} and compute the approximation error \begin{align} ||\mathcal{L} u-\mathcal{L} u^N|| = \sum_{i=N+1}^\infty \hat u_i^2 \end{align} where I ignore weights and normalization factors. We then use the definition of the expansion coefficients to proof spectral convergence \begin{align} \hat u_i = \langle u,\phi_i \rangle = \langle u,\frac{\lambda_i}{\lambda_i}\phi_i \rangle =\frac{1}{\lambda_i}\langle u,\lambda_i\phi_i \rangle = \frac{1}{\lambda_i}\langle u,\mathcal{L}\phi_i\rangle \end{align} Using integration by part and repeating this process, we end up with something like \begin{align} \hat u_i = \langle u,\phi_i \rangle \leq \frac{1}{\lambda_i^m} \langle \mathcal{L}^m u, \phi_i \rangle \leq\frac{1}{\lambda_i^m} ||u||_m \end{align} where $||\cdot ||_m$ is some appropriate norm.
Now this derivation did not use the collocation method. I fail to see, how the procedure changes when applying the collocation method instead. If I understand the method correctly, we now only requre, that the residual vanishes at the collocation points. I assume that something happens to the scalar products for $\hat u_i$ but I'm not sure how. When we use the collocation method, we compute expansion coeffiients and later on reconstruct the solution by lagrange interpolation, but how can we ensure spectral convergence?