I am going to use Lagrange interpolation using Chebyshev nodes using the following formula
$$\sum_x \prod_{k=0,k\not={j}}^n \frac {x-y_k}{y_j-y_k} f(x) $$
in which $x$ in my data points, $y_k $s are my chebyshev points. It could happen that the $x$ and $y_k$ be equal so the product will be zero. What I have to do in these situations? Should I use the zero value or put something else instead?
If $x = y_j \in X$ ($X$ Chebyshev grid) we want that all Lagrange polynomials cancels except the polynomial associated to the node $y_j$ that we want be equal to $1$.
This is the only way to verify to have: $$\Pi_nf(x) := \sum_{j = 0}^n\left( f(y_j) \prod_{k=0,k\not={j}}^n \frac {x-y_k}{y_j-y_k} \right) $$ equal to $f(y_j)$. In fact, if $x= y_j$ every term of the sum cancels except the $j$-th and so we get the result.