I\'m supposed to find the best approximation in the mean to aparticular function
ID: 2938038 • Letter: I
Question
I'm supposed to find the best approximation in the mean to aparticular function using Legendre polynomials (this is for acourse on PDEs and Fourier analysis and boundary value problems).What is the mean in the L^2 sense? Doesn't L^2 have to do withHilbert space? I don't see how that applies here. Or is it justanother name for least squares approximation? I'm supposed to find the best approximation in the mean to aparticular function using Legendre polynomials (this is for acourse on PDEs and Fourier analysis and boundary value problems).What is the mean in the L^2 sense? Doesn't L^2 have to do withHilbert space? I don't see how that applies here. Or is it justanother name for least squares approximation?Explanation / Answer
In your context it means the same as least squares. It referes to the L-2 norm which is least squares. If it was L-1 they would want to fit using the absolute value of differences. It it was L- it would be mini-max fit. Other norms may be possible, but those three are only ones I have ever seen.