# orthogonal polynomials

## 1 Orthogonal Polynomials

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

 $p_{n}(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}$

They can be differentiated and integrated for any value of $x$, and are fully determined by the $n+1$ coefficients $a_{0}\ldots a_{n}$ . 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

 $\int_{a}^{b}p_{n}(x)p_{m}(x)W(x)dx=0\;\text{for}\;n\neq m$

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 $q_{0}(x),q_{1}(x),\ldots$, 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:

$a$ $b$ $W(x)$ name
-1 1 1 Legendre polynomials
-1 1 $(1-x^{2})^{-1/2}$ Chebyshev polynomials
$-\infty$ $\infty$ $e^{-x^{2}}$ Hermite polynomials

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

 $a_{0}p_{0}+a_{1}p_{1}+\cdots+a_{n}p_{n}$

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 $x_{i}$:

 $\sum_{i}T_{n}(x_{i})T_{m}(x_{i})=0\;\text{for}\;n

where the $x_{i}$ depend on $M$.