The word approximation means near to exact but not the actual value. This is more frequently used term in our daily life since the existence of human being. In mathematical calculations, using 1.414 in place of 2 and 3.14 in place of π are the examples of approximation of an irrational or a transcendental number using a rational number. With the passage of time, this area has grown up to become approximation theory, a field of mathematical analysis.
Approximation theory deals with the approximation of functions by means of simpler functions whose behavior is already known to us. The latter category of functions is known as approximants. Approximation of real-valued functions started with the famous theorem given by “K”. Weierstrass known as Weierstrass approximation theorem which states that ‘Every real-valued continuous function defined on a closed interval [a,b] can be uniformly approximated to any degree of accuracy by a sequence of polynomials on a,b‘. For example: Taylor’s series expansion of a continuously differentiable function in a closed interval [a,b].
An important term in the approximation is what we simply call as an error, the difference between the actual value of the function and its approximation at any point. The approximation is called the best fit if the error is near to zero or insignificant. So, for a given function the aim of approximation is to find an approximant such that the error may be minimized near to zero.
Written By: Mr. Ramapati Maurya, Assistant Professor, Department of Mathematics
Manav Rachna Educational Institutions 