Suppose there are \(m\) numbers of Chebyshev nodes over a bounded interval [a,b]:
\(s_{i} \in [a,b],\) for \(i = 1,2,\cdots,m\).
These nodes can be nomralized to the standard Chebyshev nodes over the domain [-1,1]:
\(z_{i} = \frac{2(s_{i} - a)}{(b - a)} - 1\).
With normalized Chebyshev nodes, the recurrence relations of Chebyshev polynomials of order \(n\) is defined as:
\(T_{0} (z_{i}) = 1\),
\(T_{1} (z_{i}) = z_{i}\), and
\(T_{n} (z_{i}) = 2 z_{i} T_{n-1} (z_{i}) - T_{n-2} (z_{i})\).
The interpolation matrix (Vandermonde matrix) of (n-1)-th Chebyshev polynomials with \(m\) nodes,
\(\Phi_{mn}\) is:
\( \Phi_{mn} = \left[ \begin{array}{ccccc}
1 & T_{1} (z_{1}) & \cdots & T_{n-1} (z_{1})\\
1 & T_{1} (z_{2}) & \cdots & T_{n-1} (z_{2})\\
\vdots & \vdots & \ddots & \vdots\\
1 & T_{1} (z_{m}) & \cdots & T_{n-1} (z_{m})
\end{array} \right] \).
The partial derivative of the monomial basis matrix can be found by the relation:
\((1-z_{i}^{2}) T'_{n} (z_{i}) = n[ T_{n-1} (z_{i}) - z_{i} T_{n} (z_{i}) ]\).
The technical details of the monomial basis of Chebyshev polynomial can be referred from Amparo et al. (2007)
and Miranda and Fackler (2012).