Orthogonal Polynomials

  Polynomials of order n are analytic functions that can be written in the form

They can be differentiated and integrated for any value of x, and are fully determined by the n+1 coefficients a₁. For this simplicity they are frequently used to approximate more complicated or unknown functions. In approximations, the necessary order n of the polynomial is not normally defined by criteria other than the quality of the approximation.

Using polynomials as defined above tends to lead into numerical difficulties when determining the a_i, even for small values of n. It is therefore customary to stabilize results numerically by using orthogonal polynomials over an interval [a,b], defined with respect to a weight function W(x) by

Orthogonal polynomials are obtained in the following way: define the scalar product

between the functions f and g, where W(x) is a weight factor. Starting with the polynomials p₀(x)=1, p₁(x)=x, p₂(x)=x², etc., Gram-Schmidt decomposition one obtains a sequence of orthogonal polynomials such that . The normalization factors N_n are arbitrary. When all N_i are equal to one, the polynomials are called orthonormal.

Examples:

Orthogonal polynomials of successive orders can be expressed by a recurrence relation:

This relation can be used to compute a finite series

with arbitrary coefficients a_i, without computing explicitly every polynomial p_j ( Horner's Rule).

Chebyshev polynomials T_n(x) are also orthogonal with respect to discrete values x_i:

where the x_i depend on M.

also [Abramowitz74], [Press95].