Let's have $f(x)$ a nice*, scalar valued function. It is approximated by $$ f(x)\approx g(x)=\sum_i f(x_i) N_i(x), $$ where
- $x_i = x_0+i \Delta x$ are uniformly spaced points,
- $N_i(x) = N(x-x_i)$, and
- $N$ is a linear, quadratic, or cubic spline that is a partition of unity, and has finite support.
Based only this information, what can be said about bounds for $|f(x)-g(x)|$, $|f'(x)-g'(x)|$, and $|f''(x)-g''(x)|$?
What if $f$ is a degree $n$ polynomial? Can these approximations be exact?
Note:
Based on $g$'s definition $$ f'(x)\approx g'(x)=\sum_i f(x_i) N_i'(x)\\ f''(x)\approx g''(x)=\sum_i f(x_i) N_i''(x) $$
*let's say at a minimum twice continuously differentiable, but it's fine if more strict requirements are needed of the analysis.
Look in Carl deBoor’s book “A Practical Guide to Splines”. The error bounds are typically obtained by truncating Taylor series expansions. This means that the error for cubic spline approximation (for example) is related to the maximum value of the fourth derivative of $f$.