orthogonal polynomials

1 Orthogonal Polynomials

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

$$p_n(x)=a_0+a_{1}x+a_2{x}^2+\mathrm{\cdots }+a_n{x}^n$$

They can be differentiated and integrated for any value of $x$, and are fully determined by the $n+1$ coefficients $a_0\mathrm{\dots }{a}_n$ . For this simplicity they are frequently used to approximate more complicated or unknown functions. In approximations, the necessary degree $n$ 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 positive weighting function $W(x)>0$ by

$${\int }_a^b{p}_n(x)p_m(x)W(x)𝑑x=0\text{for}n\ne m$$

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

$$(f,g)={\int }_a^{b}f(x)g(x)W(x)𝑑x$$

between the functions $f$ and $g$, where $W(x)$ is a weight factor. Starting with the polynomials $p_0(x)=1$, $p_1(x)=x$, $p_2(x)=x^2$, etc., from the Gram-Schmidt decomposition one obtains a sequence of orthogonal polynomials $q_0(x),q_1(x),\mathrm{\dots }$, such that $(q_m,q_n)=N_n{\delta }_{mn}$. The normalization factors $N_n$ are arbitrary. When all $N_i$ are equal to one, the polynomials are called orthonormal.

Some important orthogonal polynomials are:

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

$$p_n=(A_n+B_{n}x)p_{n-1}+C_n{p}_{n-2}$$

This relation can be used to compute a finite series

$$a_0{p}_0+a_1{p}_1+\mathrm{\cdots }+a_n{p}_n$$

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 $M$.

For more information, see [2, 3].