Interpolation, Approximation and Galerkin Method

An interpolation of a function p is introduced when a sequence of functions is selected, and the function p is approximately represented by the sequence. For example, let us consider a function p such that 

Here e, (t = 1,2, 3, ) give a sequence of interpolation functions. Note that the sequence of functions (4.56) is referred to as the Legendre polynomials, which are orthonormal such that 

It is not necessary to use the Legendre polynomials for the approximation of the function (4.55) the following simple polynomials can also be employed  

4 Virtual Work Equation, Variational Methods and Energy Principles 

However we should note that the polynomials provided by (4.60) are not orthonormal Sij). We can use other sequences of functions provided they have a 

---