I am working on a problem where I need to produce an inner product for numerical interpolation formulas. Ideally, I would like for this inner product to satisfy an integration by parts formula that looks something like the normal IBP formula from analysis, but I am a bit stuck on how to do this, and I was wondering if anyone knew of any work that has been done in this area yet? Here is the setup for the problem:
Given an analytic function $f$, we know that we can produce the $n+1$ point finite difference approximation for the $n$-th derivative of $f$ using
$$f^{(n)}(x_0) \approx \begin{pmatrix}0 & \cdots & 0 & 1\end{pmatrix}\begin{pmatrix}1 & x_0 & \cdots & x_n\\1 & x_0^2 & \cdots & x_n^2\\\vdots & \vdots& \vdots & \vdots\\1 & x_0^n & \cdots & x_n^n\end{pmatrix}^{-1}\begin{pmatrix}f(x_0)\\f(x_1)\\\vdots\\f(x_n)\end{pmatrix}.$$
(Note that in this definition we have to assume that the sample points are all distinct). Now, the obvious way to integrate this function is just to take $\sum_{i=0}^{n-1} f^{(n)}(x_i)[x_{i + 1}-x_i] = \sum_{i=0}^{n-1} S^i[f^{(n)}(x_0)](x_{i + 1}-x_i)$ (here $S$ stands for the shift operator defined by $S[f(x_i)] = f(x_{i + 1})$ and we will assume that the points are ordered $x_0 < x_1 < \cdots < x_n$), however this produces some less than desirable results, namely, integrating $f^{(n)}$ does not produce a difference between values of $f^{(n-1)}$ except for in the case where $n = 1$.
We can get a nicer formula if we define the integral to instead be
$$\int_{x_0}^{x_{2n-1}} f^{(n)}(x_0)\,\Delta x = \sum_{i=0}^{n-1} f^{(n)}(x_i)[x_{n + i}-x_i] = \sum_{i=0}^{n-1} S^i[f^{(n)}(x_0)](x_{n + i}-x_i)$$
(I have abused the notation slightly here. When I say $f^{(n)}(x_0)$, I actually mean it's finite difference interpolation formula with lowest value $x_0$) in which case, we can get that the integral behaves, more or less, as we would expect an integral to behave. Namely, we have that $$\int_{x_0}^{x_{2n-1}}f^{(n)}(x_0)\,\Delta x = f^{(n-1)}(x_n) - f^{(n-1)}(x_0).$$
With this, it seems that we have found a plausible candidate for the inner product $\langle f^{(n)}, 1\rangle$, but when I try to expand this to something like $\langle f^{(n)},g^{(m)}\rangle$, everything gets really messy, so I figured before I walked down this nightmarish road of algebra, I would quickly see if anyone else knew if this has already been done before and could provide me with a reference? I have already done quite a bit of digging in my old numerics books and on MathSciNet, but I haven't been able to locate anything useful. Thanks in advance for the help!