Orthogonal polynomials

Orthogonal polynomials are a set of polynomial functions that are mathematically independent of each other in a specific way. When we say they are "orthogonal," we mean that the product of any two different polynomials from the set, integrated over a certain interval with a specific weight, equals zero. This property makes them useful in approximation, solving differential equations, and data analysis, because they simplify complex problems by breaking them into simpler, non-overlapping components. Think of them as a special coordinate system that helps us analyze and approximate functions more efficiently.