Overview

Function approximation takes a limited set of information about a function, such a set of points, or a set of derivatives, and tries to approximate the function from that information. Most of the techniques are based on a functional analysis foundation.

Formal Definition

The formal definition of an approximation often assumes a normed function space {% K %}, (a Banach Space) and a subset of that space, representing the set of approximations. The best approximation is then taken to be the closest point

Approximation by Series

• Taylor Series : is a well known calculus technique for approximating a function using a polynomial and set of known derivatives at a given point.
• Legendre Polynomials :
• Fourier Series :

Other Approximation Techniques

• Interpolation Techniques : take a set of points, sampled from the functions domain with the corresponding function outputs, and tries to create a function that fits the sample points.
• Optimization : takes a given functional form (that is, a function specified by a set of unknown parameters), and finds the closest fit to a set of points by tuning the parameters that the function depends upon.
• Machine Learning :