Consider a discrete function $f : \{x_1,\dots,x_n\} \to [a,b]$ that, for some $\omega>0$, satisfies
$$\max_i |f_i-f_{i-1}| \le \omega. \tag1$$
Let $q$ be the polynomial of order $m<n$ that, for some $p$, approximates the function $f$ by minimizing
$$D_p\equiv||f-q||_p = \left(\sum_{i=1}^n|f(i) - q(i)|^p\right)^{\frac{1}{p}}.$$
Is there a sharp or a "relatively tight" upper bound to $D_p$ that does not depend on the values of $f$?
I thought that this would be easy to find online but all approximation results I could find (like Jackson's inequality) are for continuous functions. One would think that tighter bounds could be established for discrete functions, but I haven't been able to find one. Any suggestion in this direction would be very welcome.
Edit based on the comments by @Ian
Let $X$ be a matrix with elements $X_{ij}=x_i^j$, for $i=\{1,\dots,n\}$ and $j=\{0,\dots,m\}$, and let $y=[f_1,\dots,f_n]^T$. Then, for $p=2$ it follows from the least squares formula that $$D_2=||(I_n-X(X^T X)^{-1}X^T)y||_2.$$ It would also follow that, for all $p$ $$D_p\leq ||(I_n-X(X^T X)^{-1}X^T)y||_p.$$ So, one way to get a relatively tight bound (I haven't figure out how to formalize this) would be to use condition $(1)$ to bound $D_2$.