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Approximation theory asks how well members of a chosen class can represent a target function, using a specified measure of error. For a norm $\Vert \cdot \Vert$ and an approximation space $V_n$, a fundamental quantity is ${\inf }_{p\in V_n}\Vert f-p\Vert$. The choice of norm distinguishes controlling the worst error from controlling an average squared error.

For $f(x)=x^2$ on $[-1,1]$, approximate by a constant $c$. The uniform norm error is $\max (|c|,|1-c|)$, minimized at $c=1/2$ with error $1/2$. Under the squared L2 space error, differentiating ${\int }_{-1}^1(x^2-c{)}^{2}dx$ gives the different optimum $c=1/3$. Neither constant is universally best: each solves a different problem.

Polynomial approximation, Fourier series and splines provide different approximation classes. Smoothness of the target controls attainable rates, while numerical conditioning controls whether a representation can be computed reliably. A small best-approximation error alone does not guarantee a stable algorithm.

Carl de Boor authored notes on approximation theory developing best approximation, B-splines and spline interpolation. The notes connect concrete recurrence formulas with structural criteria such as the Schoenberg–Whitney theorem. This makes it possible to analyze when a spline representation exists and is unique, rather than relying only on numerical fitting.^[ref]