## Polynomial L^{2} Approximation »

### Relating Orthonormal Polynomials, Gram—Schmidt Orthonormalization, QR Factorization, Normal Equations and Vandermonde and Hilbert Matrices

This didactic synthesis compares three solution methods for polynomial approximation and systematically presents their common characteristics and their close interrelations:

1. Classical Gram–Schmidt orthonormalization and Fourier approximation in

2. Linear least-squares solution via QR factorization on an equally spaced grid in

3. Linear least-squares solution via the normal equations method in and on an equally

spaced grid in

The first two methods are linear least-squares systems with Vandermonde matrices ; the normal equations contain matrices of Hilbert type . The solutions on equally spaced grids in converge to the solutions in All solution characteristics and their relations are illustrated by symbolic or numeric examples and graphs. Read More »