Lagrange Polynomial

Lately we have been working on different methods of numerical approximation for periodic and non-periodic functions from a certain interval.  It started off by working with the idea of a Fourier Transform of a function and using it to approximate the line,  however this method would only work for us, to our desired accuracy, if we had a function that was periodic.  When we tried to use our  Fourier Transform on a non-periodic function we were met with unsatisfactory results which included unwanted jumps near the boundaries of the interval.  This led us to seek a different way to approach the function that instead of using  the Fourier method, which uses periodic functions like Sine and Cosine, but to use a different kind of functions altogether.  Then we were introduced to the Lagrange Polynomial,  an interpolation technique which uses a polynomial basis to approximate the function.

The Lagrange polynomial is a very powerful technique which creates