Orthogonal polynomials¶

This section provides some mathematical details on sequences of orthogonal polynomials. Some of these sequences will be used to construct the basis of the so-called polynomial chaos expansion.

Mathematical framework

The orthogonal polynomials are associated to an inner product, defined as follows:

Given an interval of orthogonality $[\alpha,\beta]$ ( $\alpha \in \Rset \cup \{-\infty\}$ , $\beta \in \Rset \cup \{\infty\}$ , $\alpha < \beta$ ) and a weight function $w(x)> 0$ , every pair of polynomials $P$ and $Q$ are orthogonal if:

$\langle P,Q \rangle = \int_{\alpha}^{\beta}P(x)Q(x)~w(x) dx = 0$

Therefore, a sequence of orthogonal polynomials $(P_n)_{n\geq 0}$ ( $P_n$ : polynomial of degree $n$ ) verifies:

$\langle P_m,P_n\rangle = 0 \text{~~for every~~} m \neq n$

The chosen inner product induces a norm on polynomials in the usual way:

$\parallel P_n\parallel=\langle P_n,P_n \rangle^{1/2}$

In the following, we consider weight functions $w(x)$ corresponding to probability density functions, which satisfy:

$\int_{\alpha}^{\beta} \; w(x) \; dx \, \, = \,\, 1$

Moreover, we consider families of orthonormal polynomials $(P_n)_{n\geq 0}$ , that is polynomials with a unit norm:

$\|P_n\| \, \, = \, \, 1$

Any sequence of orthogonal polynomials has a recurrence formula relating any three consecutive polynomials as follows:

$P_{n+1}\ =\ (a_nx+b_n)\ P_n\ +\ c_n\ P_{n-1}$

Orthogonormal polynomials with respect to usual probability distributions

Below, a table showing an example of particular (normalized) orthogonal polynomials associated with continuous weight functions.

Note that the orthonormal polynomials are orthonormal with respect to the standard representative distribution of the given distribution.

Ortho. poly.	$P_n(x)$	Weight $w(x)^{\strut}$	Recurrence coefficients $(a_n,b_n,c_n)$
Hermite	${He}_n(x)^{\strut}$	$\displaystyle \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}}$	$\begin{array}{ccc} a_n & = & \frac{1}{\sqrt{n+1}} \\ b_n & = & 0 \\ c_n & = & - \sqrt{\frac{n}{n+1}} \end{array}$
Legendre	$\begin{array}{c} {Le}_n(x) \\ \\ \alpha>-1 \\ \end{array}$	$\displaystyle \frac{1}{2}^{\strut} \times \mathbb{I}_{[-1,1]}(x)$	$\begin{array}{ccc} a_n & = & \frac{\sqrt{(2n+1)(2n+3)}}{n+1} \\ b_n & = & 0 \\ c_n & = & -\frac{ n \sqrt{2n+3} }{ (n+1)\sqrt{2n-1} } \end{array}$
Laguerre	$L_n^{(\alpha)}(x)$	$\displaystyle \frac{x^{k-1}}{\Gamma(k)}~e^{-x} \mathbb{I}_{[0,+\infty[}(x)$	$\begin{array}{ccc} \omega_{n} & = & \left((n+1)(n+k+1) \right)^{-1/2} \\ a_n & = & \omega_{n} \\ b_n & = & -(2n+k+1)~\omega_{n} \\ c_n & = & -\sqrt{(n+k)n}~\omega_{n} \end{array}$
Jacobi	$\begin{array}{c} J^{(\alpha,\beta)}_n(x) \\ \\ \\ \alpha,\beta>-1 \\ \end{array}$ \| $\frac{(1-x)^{\alpha}(1+x)^{\beta}}{2^{\alpha + \beta + 1} B(\beta + 1, \alpha + 1)} \mathbb{I}_{[-1,1]}(x)$		$\begin{array}{ccc} K_{1,n} & = & \frac{2n+\alpha + \beta + 3}{(n+1)(n+\alpha+1)(n+\beta+1)(n+\alpha+\beta+1)} \\ \\ K_{2,n} & = & \frac{1}{2} \sqrt{(2n + \alpha + \beta + 1) K_{1,n}} \\ \\a_n & = & K_{2,n}(2n+\alpha + \beta + 2) \\ \\ b_n & = & K_{2,n}\frac{(\alpha - \beta)(\alpha + \beta)}{2n+\alpha+\beta} \\ \\ c_n & = & - \frac{2n+\alpha+\beta + 2}{2n+\alpha+\beta} \Big[(n+\alpha)(n+\beta) \\ & & \times (n+\alpha+\beta)n\frac{K_{1,n}}{2n+\alpha+\beta-1}\Big]^{1/2} \end{array}$

Furthermore, two families of orthonormal polynomials with respect to discrete probability distributions are available, namely the so-called Charlier and Krawtchouk polynomials:

Ortho. poly.	$P_n(x)$	Probability mass function	Recurrence coefficients $(a_n,b_n,c_n)$
Charlier	$\begin{array}{c} Ch^{(\lambda)}_n(x) \\ \\ \lambda>0 \\ \end{array}$	$\begin{array}{c} \displaystyle{\frac{\lambda^k}{k!}~e^{-\lambda}} \\ \\ k=0,1,2,\dots \\ \end{array}$	$\begin{array}{ccc} a_n & = & - \frac{1}{\sqrt{\lambda (n+1)}} \\ \\ b_n & = & \frac{n+\lambda}{\sqrt{\lambda (n+1)}} \\ \\ c_n & = & - \sqrt{1 - \frac{1}{n+1}} \end{array}$
Krawtchouk $^{\dagger}$	$\begin{array}{c} Kr^{(m,p)}_n(x) \\ \\ m \in \Nset~,~p \in [0,1] \\ \end{array}$	$\begin{array}{c} \displaystyle{\binom{m}{k}p^k (1-p)^{m-k}} \\ \\ k=0,1,2,\dots \\ \end{array}$	$\begin{array}{ccc} a_n & = & - \frac{1}{\sqrt{(n+1)(m-n)p(1-p)}} \\ \\ b_n & = & \frac{p(m-n)+n(1-p)}{\sqrt{(n+1)(m-n)p(1-p)}} \\ \\ c_n & = & - \sqrt{(1 - \frac{1}{n+1})(1+\frac{1}{m-n})} \end{array}$

$^{\dagger}$ The Krawtchouk polynomials are only defined up to a degree $n$ equal to $m-1$ . Indeed, for $n=m$ , some factors of the denominators of the recurrence coefficients would be equal to zero.

To sum up, the distribution type are reported in the table below together with the associated families of orthonormal polynomials.

Distribution	Support	Polynomial
Normal $\cN(0,1)$	$\Rset$	Hermite
Uniform $\cU(-1,1)$	$[-1,1]$	Legendre
Gamma $\Gamma(k,1,0)$	$(0,+\infty)$	Laguerre
Beta $B(\alpha,\beta,-1,1)$	$(-1,1)$	Jacobi
Poisson $\cP(\lambda)$	$\Nset$	Charlier
Binomial $\cB(m,p)$	$\{0,\dots,m\}$	Krawtchouk $^{\dagger}$

$^{\dagger}$ It is recalled that the Krawtchouk polynomials are only defined up to a degree $n$ equal to $m-1$ .

Orthogonal polynomials with respect to arbitrary probability distributions

It is also possible to generate a family of orthonormal polynomials with respect to an arbitrary probability distribution $w(x)$ . The well-known Gram-Schmidt algorithm can be used to this end. Note that this algorithm gives a constructive proof of the existence of orthonormal bases.

However it is known to be numerically unstable, so alternative procedures are often used in practice. The available orthonormalization algorithm is the Stieltjes algorithm.

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Orthogonal polynomials¶