Math StackExchange, Wikipedia and Proof Wiki all offer roughly similar proofs of the error formula for polynomial interpolation of a $n+1$ differentiable function $f(x)$ with a $n$-degree polynomial $p_n(x)$:

$$f(x) - p_n(x) = \frac{f^{n+1}(t)}{(n+1)!} \prod_{i=0}^n (x - x_i)$$

The proofs proceed by constructing a new function that has n+2 roots and taking n+1 derivatives, and I can follow the proofs well enough. For example, the Math StackExchange link above defines the auxiliary function as $F(x) = f(x) - p(x) - (f(\bar{x}) - p_n(\bar{x}))(\prod_{i=0}^n (x-x_i))^{-1}(\prod_{i=0}^n (x-x_i))$.

I have two questions:

  1. What motivates this auxiliary function? It appears that we're trying to give $F(x)$ an additional root, but why is this the correct starting step?
  2. The error formula appears unintuitive to me. It links the interpolation error to a many-time-differentiated function multiplied by a polynomial over a factorial. How can I easily see why this error statement is true? What's a good way to explain why the interpolation error equals the right hand side?


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