I am numerical analyst, and want to prove a result I observe numerically.
Let $f$ be a periodic function on $I = [-\pi,\,\pi]$ that is $q$ times continuously differentiable, with the $(q+1)$th derivative of bounded variation. The function $f$ satisfies $f(x_{i}) = g(x_{i})$ on the uniformly distributed discrete points $a \leq x_{1} < x_{2} < \ldots < x_{P}\leq b$ where $J = [a,\,b]$ is a proper subset of $I$. We may assume that $g$ has the same regularity properties as $f$.
Approximate f as a truncated Fourier series $s_{N}(x) = \sum\limits_{n = -N/2}^{N/2} c_{n}e^{inx}$. Then by e.g. [1] the Fourier coefficients $\{c_{n}\}$ decay as $\mathcal{O}(N^{-q-2})$. We also have $s_{N}(x_{i})-g(x_{i}) = 0$ for $i = 1,\ldots,P$.
Now to the question: I want to prove that $\max\limits_{x\in J}(|s_N(x) - g(x)|)$ decays as $\mathcal{O}(P^{-q-2})$, as $P$ and $N$ go to infinity. Here we assume that $N$ in the Fourier series goes to infinity at the same rate as $P$. (Numerically I have a disctretization of $I$ consisting of $N+1$ uniformly distributed points, with the uniformly distributed points $\{x_{i}\}$ being a subset of those.)
Ideas: I can't apply the Riemann localization theorem, since $f$ is not equal to $g$ in a neighborhood of each $x_{i}$, but only at the discrete points $\{x_{i}\}$. I have tried to simply study $h = f-g$ and bound the maxnorm of $h$ over $J$, but then I only get that the maxnorm decays as $\mathcal{O}(P^{-q})$.
References: [1] L. N. Trefethen, Spectral methods in MATLAB, https://epubs.siam.org/doi/book/10.1137/1.9780898719598