Polynomials, sampling theory, signal processing, Fourier series, etc. are all so closely related that learning even a little bit of one of these things opens up such a wealth of other things to think about and explore. Splines like this basically answer the question "what polynomial fits these points the best?" which, it turns out, is the same question you need to pose to begin formulating gaussian quadrature. And gaussian quadrature is super cool because it approximates integrals about a billion times better than riemann summation[1] for the same amount of computational effort.
I'm taking a numerical algorithm class and have been learning about gaussian quadrature for the first time. It's super cool stuff. I haven't been able to wrap my head around why it works entirely and have found readings about legendre polynomials really difficult to understand. Do you have any intuition you can share?
It's by no means rigorous, but the gist of it is that
1. You're integrating the polynomial that best approximates the function over the interval. This is why quadrature is exact for polynomials up to some order: the best approximating polynomial is the polynomial itself.
2. Polynomials are good approximations when the function is smooth. Most useful functions are.
3. Because of the smoothness, the behavior in the middle of the interval is largely affected by the behavior at the edges, so you need more densely sample the edges. Think of it like wiggling a string. You're only allowed to wiggle the end, but that very specifically defines the behavior in the middle.
4. There are lots of polynomial sets - Legendre, Chebyshev, Hermite, etc. They're each useful because they're orthogonal to a special weight function, and Legendre polynomials are kind of the default set because they have the simplest weight function: 1.
[1]: https://www.youtube.com/watch?v=k-yUdqRXijo