Module 3 : Problem Discretization using Approximation Theory
Section 2 : Polynomial Approximation[3]
 
Theorem 6 (Weierstrass Approximation Theorem):
Consider space $C[a,b]$, the set of all continuous functions over interval $[a,b]$, together with $\infty -$norm defined on it as MATH                            --------- (7)Given any $\varepsilon >0,\,$for every $f(t)\in C[a,b]$ there exists a polynomial $p_{n}(t)$ such that MATH

This fundamental result forms the basis of the problem discretization in majority of the cases. It may be noted that this is only an existence theorem and does not provide any method of constructing a polynomial approximation. The following three approaches are mainly used for constructing approximating polynomials:

  • Taylor series expansion
  • Polynomial interpolation
  • Least square approximation

These approaches and their applications to problem discretization will be discussed in detail in the sections that follow.