Context
I'm a PhD student in Statistics and I have evaluations of a $L_2([0,1])$ function $f$, that is $m$ times derivable, on a regular grid of $[0,1]$ $$f\left(\frac{k}{p-1}\right), 0\leq k \leq p-1.$$
Goal
My goal is to reconstruct $f$ on $[0,1]$ in its entirety with a good $L_2([0,1])$ approximation.
And especially to have an upper-bound on how good the reconstruction is in the sense of the $L_2([0,1])$-norm (especially in regards to the number of observation points $p$ and the smoothness of the function $m$).
A path to a solution
A usual solution in approximation of functions is to project the function on a suitable basis of $L_2([0,1])$.
For example, letting ${\Lambda_D}$ be a set of indices of finite size $D>0$ and $\left(\phi_\lambda\right)_{\lambda\in\Lambda_D}$ a set of wavelets.
We can then define $P_{\Lambda_D}$ to be the $L_2([0,1])$-orthogonal projector onto $span\left( \phi_\lambda,\lambda\in\Lambda \right)$ such that $$P_{\Lambda_D} f:=\sum_{\lambda=\Lambda_D}\left \langle f,\phi_\lambda \right \rangle_{L_2([0,1])} \phi_\lambda.$$
There exists some results about the upper-bound of $\left \Vert f-P_{\Lambda_D}f \right \Vert_{L_2([0,1])}$ expressed in relationship to $p$ and $m$ which seemed promising to me (see for example Theorem 5 of this course, or in the renowned book from R. Devore and G. Lorentz Constructive Approximation).
Problem
The problem is that I only access $f$ through a regular grid, I can not compute the scalar products required to compute $P_{\Lambda_D} f$...
I can replace the scalar products by quadratures of scalar products $Q_\lambda(f)\approx \left \langle f,\phi_\lambda \right \rangle_{L_2([0,1])}$ and set $$R_{Q,\Lambda_D} f:=\sum_{\lambda=\Lambda_D}Q_\lambda(f) \phi_\lambda.$$ But I can not manage to have a good control over $\Vert R_{Q,\Lambda_D}f - P_{\Lambda_D}f\Vert_{L_2([0,1])}$...
I'm sure this is a problem that other people have had to deal with, and I've tried to find out if there are any articles that answer this question, but I'm having a really hard time finding my way around...