> Lagrange interpolations are only distantly related to Legendre/Hermite/etc polynomials
Actually ... take an nth degree polynomial from your favorite orthogonal set. For each of its n+1 zeros, construct by Lagrange interpolation the polynomial which is 1 at that zero and 0 at the others. Then that set of n+1 interpolating polynomials has the same relation to the orthogonal polynomials of degree 0 through n that (periodized) sincs have to sinusoids, i.e. you can decompose an n degree or lower polynomial in either basis, and the two representations are related by something like a Fourier transform.
Huh, cool. I was familiar with using the division algorithm to prove that Gauss quadrature worked, but I never chased the notion any further. I suppose you're right, the DCT does the exact same thing when it samples a function at a finite number of points.
Actually ... take an nth degree polynomial from your favorite orthogonal set. For each of its n+1 zeros, construct by Lagrange interpolation the polynomial which is 1 at that zero and 0 at the others. Then that set of n+1 interpolating polynomials has the same relation to the orthogonal polynomials of degree 0 through n that (periodized) sincs have to sinusoids, i.e. you can decompose an n degree or lower polynomial in either basis, and the two representations are related by something like a Fourier transform.