I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials.

*Pointwise Lagrange interpolation*

Given a function $f \in C^0([-1,1])$ and a grid of $n+1$ nodes $X = (x_i)_{i = 0}^n$ on $[-1,1]$ we can construct the interpolating polynomial $\Pi^{\ \small{X}}_{\ n} f \in \mathbb{P}_n$ in the following way: $$\Pi^{\ \small{X}}_{\ n} f := \sum_{k = 0}^n f(x_k) L_k(x)$$ with $L_k$ the $k$-th Lagrange polynomial: $L_k(x) := \prod_{j = 0, j \not = k}^n \frac{x - x_j}{x_k - x_j} $

It's easy to check that the operator $\Pi^{\ \small{X}}_{\ n} : C^0([-1,1]) \to \mathbb{P}_n, \ f \mapsto \Pi^{\ \small{X}}_{\ n} f$ is a projection, i.e. $(\Pi^{\ \small{X}}_{\ n})^2 = \Pi^{\ \small{X}}_{\ n}$ and $\Pi^{\ \small{X}}_{\ n}|_{\mathbb{P}_n} = \mathrm{id}_{\mathbb{P}_n}$.

In order to achieve good convergence and to avoid Runge phenomenon, we can choose Chebyshev-Gauss (CG) grid (i.e. zeros of Chebyshev polynomials) getting the interpolating polynomial: $\Pi^{\ \small{CG}}_{\ n} f$. We can make also another choice and take Chebyshev-Gauss-Lobatto grid (CGL or simply GL) obtaining $\Pi^{\ \small{GL}}_{\ n} f$

*Chebyshev Expansion*

In a more abstract framework, we can start by considering the space $L^2_{\small w}([-1,1])$ with the following inner product $(f,g)_w := \int_{-1}^1 f(x)g(x)w(x)dx$. Then , choosing $w(x) = 1/\sqrt{1 - x^2}$, we obtain the Chebyshev orthonogonal basis $\{T_k\}_{k\ge0}$ where $T_k$ is the $k$-th Chebyshev polynomial.

We can write:$$f(x) = \sum_{k=0}^\infty \hat{f}_k T_k(x) \quad \forall x \in [-1,1]$$ with $\hat{f}_k = (f,T_k)_w/(T_k,T_k)_w$

Moreover we can truncate the series at $n$ and obtain a projection of $f$ onto $\mathbb{P}_n$: $$S_{\ n} f := \sum_{k=0}^n \hat{f}_k T_k$$

*Connections?*

My question is this: is possible to find coefficients $\hat{f}_k$ such that: $$ \Pi^{\ \small{CG}}_{\ n} f = S_{\ n} f = \sum_{k=0}^n \hat{f}_k T_k$$ or $$ \Pi^{\ \small{GL}}_{\ n} f = S_{\ n} f = \sum_{k=0}^n \hat{f}_k T_k$$

Furthermore I know that we can approximate coefficients $\hat{f}_k$ using a discrete scalar product i.e. approximating the integral with interpolating polynomial but It seems a closed loop...

Can anyone help me?

Note: *I originally posted this question on math.se but didn't receive any answer (also starting a bounty). I hope this question can be appropriate for this site, I'm sorry in advance if this is not "research level" math.*