In mathematics, approximation theory is concerned with how functions can be best approximated with simpler functions, and with quantitatively characterising the errors introduced thereby. What is meant by best and simpler will depend on the application.
Approximation theory has many applications, especially in numerical computation, physics, engineering and computer science. There are two main applications of approximations. The first is approximating safisticated functions in a computer mathematical library, using simpler operations that can be performed on the computer (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or rational approximations. The second is obtaining approximate values of real-world function (known on the grid) between the points of the grid.