I am studying spectral methods for numerical solutions for PDEs. Currently, I am on a chapter that explains how to use Chebyshev polynomials to solve non-periodic boundary value problems.
I understand that we should scale our domain to $x \in [-1, 1]$. After that, we should define our approximate solution to be the polynomial $p_n(x)$. Also, this polynomial satisfies the condition:
$$ p_n(x_j)=v(x_j) \ \forall \ j = 0,1,2...N\tag 1$$
where $v(x)$ is the exact solution to our PDE, and $x_j$ are the Chebyshev's collocation points defined by:
$$ x_j = \cos \Bigg({j \pi \over N}\Bigg) \tag 2$$
However, I don't understand how exactly is the $p_n(x)$ analytically defined. Is it of the form:
$$ p_n(x) = \sum_{j=0}^N a_jT_j(x) \tag 3$$
where $a_j$ are the weight factors that should be determined using the spectral method, or is it of some other form? This is not really clear to me from literature.