I know the mathematical definition but I'm having a hard time understanding the utility of orthogonal polynomials. I'm not saying they are useless, far from that! It is just that I like understanding thinks from a higher level than its mathematical definition...
- What are the interesting properties of this type of polynomials?
- Why is it relevant that their inner product is equal to 0?
- What is the beauty on them?
To provide some context, I'm currently working with polynomial chaos expansions and I have to explain the method to non mathematicians. I need to be able to explain it in simple words. This is why I need to understand orthogonal polynomials in simple words
Such families of orthogonal polynomials allow numerical quadratures (for example Gauss-Legendre-quadrature).
Another application is the approximation of a function by tchebycheff-polynomials.
The fact, that the inner product is zero helps to find simple formulas for the mentioned applications.
Also, these polynomials have some interesting minimality properties used to interpolate a function.