Legendre Polynomials

Background

    We have seen how Newton polynomials and Lagrange polynomials are used to approximate on an interval .  The constructions were based on a discrete set of interpolation points in the interval.  We will now consider "least squares" approximations where the polynomial is "close" to the function throughout the interval .  Our final construction will use Legendre polynomials that were first studied by the French mathematician Adrien-Marie Legendre (1752-1833).
    
    Given a set of interpolation points the Newton polynomial and Lagrange polynomial are algebraically equivalent, and they are equivalent to the polynomial constructed with Mathematica's built in subroutine "InterpolatingPolynomial."  The subroutine "Fit" can be used to construct the discrete least squares fit polynomial.

Definition (Discrete Least Squares Approximation)  Given a function   on  and equally spaced nodes   and interpolation points .  The degree polynomial    is the discrete least squares interpolation fit provided that the coefficients    of    minimize the sum

            

Theorem (Discrete Least Squares Approximation)  The polynomial    satisfies the equations  
    
                for   .  

These equations can be simplified to obtain the normal equations for finding the coefficients   

                 for   .   

Remark. This is the degenerate case of a least squares fit (i.e. if there were   data points we would have used    instead of  ).
Information on polynomial curve fitting can be found in the module Least Squares Polynomials.

    The following example shows that if n+1 points are used to find the discrete least squares approximation polynomial of degree n , then it is the same as the Newton (and Lagrange) interpolation polynomial that passes through the n+1 points.  

Theorem (Continuous Least Squares Approximation)  The polynomial    satisfies the equations  
    
                for   .  

These equations can be simplified to obtain the normal equations for finding the coefficients   

                 for   .  

Mathematica Function ( Inner Product ). To compute the inner product of two real functions over .  

Proof  Legendre Polynomials  

Efficient Computations

    We now present the efficient way to compute the continuous least squares approximation.  It has an additional feature that each successive term increases the degree of approximation.  Hence, an increasing sequence of of approximations can obtained recursively:  
    
          
        
Theorem (Legendre Series Approximation)  The Legendre series approximation of order for a function over    is given by
    
        

where    is the Legendre polynomial and

        

The Shifted Legendre Polynomials

    The "shifted Legendre polynomials
    
          are orthogonal on ,  
    
    where are the Legendre polynomials on .