= Approximation theory
<Approximation theory> asks how well members of a chosen class can represent a target <function>, using a specified measure of error. For a <norm> $\|\cdot\|$ and an approximation space $V_n$, a fundamental quantity is $\inf_{p\in V_n}\|f-p\|$. 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.
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