# Function approximation

In general, a function approximation problem asks us to select a function among a well-defined class that closely matches ("approximates") a target function in a task-specific way. The need for **function approximations** arises in many branches of applied mathematics, and computer science in particular.

One can distinguish two major classes of function approximation problems:

First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).

Second, the target function, call it *g*, may be unknown; instead of an explicit formula, only a set of points of the form (*x*, *g*(*x*)) is provided. Depending on the structure of the domain and codomain of *g*, several techniques for approximating *g* may be applicable. For example, if *g* is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. If the codomain (range or target set) of *g* is a finite set, one is dealing with a classification problem instead.

To some extent, the different problems (regression, classification, fitness approximation) have received a unified treatment in statistical learning theory, where they are viewed as supervised learning problems.

## See also

- Approximation theory
- Kriging
- Least squares (function approximation)
- Radial basis function network
- Fitness approximation