Approximation theory asks how well members of a chosen class can represent a target function, using a specified measure of error. For a norm and an approximation space , a fundamental quantity is . The choice of norm distinguishes controlling the worst error from controlling an average squared error.
For on , approximate by a constant . The uniform norm error is , minimized at with error . Under the squared L2 space error, differentiating gives the different optimum . 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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Approximation theory is a branch of mathematics that focuses on how functions can be approximated by simpler or more easily computable functions. It deals with the study of how to represent complex functions in terms of simpler ones and how to quantify the difference between the original function and its approximation. The field has applications in various areas, including numerical analysis, functional analysis, statistics, and machine learning, among others.