I have read Philipp Schlatter's Lecture Notes: Spectral Methods in 2010. A hyperlink is provided below.
Lecture Notes: Spectral Methods
At Page 2, Eq. $(3)$ in this notes, the author have written the spatial derivative as follows:
$$ \frac{\partial^p u_N}{\partial x^p}=\sum_{k=0}^{N} a_k(t)\cdot\frac{d^p}{dx^p}\phi_k(x)=\sum_{k=0}^{N'}a_k^{(p)}(t)\cdot\phi_k(x)\,. $$
Note that $\cdot$ is just scalar multiplication (scalar product in $1$D) and $u_N$ is an approximate solution to an IBVP, $P[u]=0$. The author stated that the derivative was decoupled using the finite number of trial functions $\phi_k(x)$, and I agree with this as the basic content. However, I don't understand how the second equality in the above equation holds. Furthermore, there is no definition of $a_k^{(p)}$ in the notes, I just guess it is a derivative with respect to time as a common. It does not deal with specific partial differential equations, so it is accompanied by a very unclear understanding. I am curious as to why the above second equality is established.
After exploring a little deeper, the following conclusions are drawn. If the trial functions $\phi_k(x)$ are polynomials with the larget order $n<\infty$, one can set all the coefficients to $0$ for $\phi_k(x)$ whose order exceeds $n-p$, which are represented by $a_k^{(p)}$ in the notes. Also, if the ansatz functions are periodic functions, $u_N$ can be uniquely represented by changing the coefficients, $a_k(t)$ to $a_k^{(p)}(t)$. However, I think that it would be better if the trial functions in RHS were expressed in other notation such as $\psi_k(x)$ instead of $\phi_k(x)$. Furthermore, $a_k^{(p)}$ must be denoted in the context, since this superscript implies the high-order derivative in a common.