Mathematical Background

This section is heavily based on the book "Orthogonal Polynomials: Computation and Approximation" by Walter Gautschi (Oxford University Press)

Orthogonal Polynomials

Basic Theory

We always work with absolutely continuous measures for which we write $\mathrm{d} \lambda (t) = w(t) \mathrm{d}t$, where the so-called weight function $w$

  • is a nonnegative integrable function on the real line $\mathbb{R}$, i.e. $w: \mathcal{W} \subseteq \mathbb{R} \rightarrow \mathbb{R}_{\geq 0},$
  • has finite limits in case $\mathcal{W} = \mathbb{R}$, i.e.

\[\lim_{t \to \pm \infty} w(t) < \infty,\]

  • has finite moments of all orders

\[\mu_r(\mathrm{d}\lambda) = \int_{\mathcal{W}} t^r \mathrm{d} \lambda (t), \quad r = 0, 1, 2, \dots \quad \text{with}\: \mu_0 > 0.\]

For any pair of integrable functions $u, v$, their scalar product relative to $\mathrm{d} \lambda$ is defined as

\[\langle u, v \rangle_{\mathrm{d} \lambda} = \int_{\mathcal{W}} u(t) v(t) \mathrm{d} \lambda(t)\]

Let $\mathcal{P}$ be the set of real polynomials and $\mathcal{P}_d \subset \mathcal{P}$ be the set of real polynomials of degree at most $d$ on $\mathcal{W}$, respectively. Monic real polynomials are real polynomials with leading coefficient equal to one, i.e. $\pi_k(t) = t^k + \dots$ for $k = 0, 1, \dots.$

The polynomials $u,v \in \mathcal{P}$ with $u \neq v$ are said to be orthogonal if

\[\langle u, v \rangle_{\mathrm{d} \lambda} = \int_{\mathcal{W}} u(t) v(t) \mathrm{d} \lambda(t) = 0.\]

The norm of $u$ is given by

\[\| u \|_ {\mathrm{d}\lambda} = \sqrt{\langle u, u \rangle}.\]

If the polynomials $u \in \mathcal{P}$ has unit length $\| u \|_ {\mathrm{d}\lambda} = 1$, it is called orthonormal.

Monic orthogonal polynomials are polynomials that are monic and orthogonal, hence satisfy

  • \[\pi_k(t) = \pi_k(t; \mathrm{d} \lambda) = t^k + \dots\]

    for $k = 0, 1, \dots$, and
  • \[\langle \pi_k, \pi_l \rangle_{\mathrm{d}\lambda} = 0\]

    for $k \neq l$ and $k, l = 0, 1, \dots$, and $\langle \pi_k, \pi_k \rangle_{\mathrm{d}\lambda} = \| \pi_k \|^2_ {\mathrm{d}\lambda} > 0$ for $k = 0, 1, \dots$.
Note

The support $\mathcal{W}$ of $\mathrm{d} \lambda$ can be an interval (finite, half-finite, infinite), or a finite number of disjoint intervals. If the support consists of a finite or denumerably infinite number of distinct points $t_k$ at which $\lambda$ has positive jumps $w_k$, the measure is called a discrete measure. For a finite number $N$ of points, the discrete measure is denoted by $\mathrm{d}\lambda_N$, and it is characterized by its nodes and weights $\{ t_k, w_k \}_{k=1}^N$ according to

\[\mathrm{d} \lambda_N (t) = \sum_{k=1}^N w_k \delta(t-t_k),\]

where $\delta$ is the $\delta$-function.

The inner product associated with $\mathrm{d} \lambda_N$ is

\[\langle u, v \rangle_{\mathrm{d}\lambda_N} = \int_{\mathcal{W}} u(t) v(t) \mathrm{d} \lambda_N (t) = \sum_{k=1}^{N} w_k u(t_k) v(t_k).\]

There exist only $N$ orthogonal polynomials $\{ \pi_k(\mathrm{d} \lambda_N) \}_{k=0}^{N-1}$ that are orthogonal relative to the discrete measure $\mathrm{d} \lambda_N$ in the sense

\[\langle \pi_k(\mathrm{d} \lambda_N) \pi_l(\mathrm{d} \lambda_N) \rangle_{\mathrm{d}\lambda_N} = \| \pi_k(\mathrm{d} \lambda_N) \|_{\mathrm{d} \lambda_N} \delta_{kl},\]

where $\delta_{kl}$ is the Dirac-delta, for $k,l = 0, 1, \dots, N-1$.

Properties

Symmetry

An absolutely continuous measure $\mathrm{d} \lambda(t) = w(t) \mathrm{d} t$ is symmetric (with respect to the origin) if its support is $\mathcal{W} = [-a,a]$ for some $0 < a \leq \infty$, and if $w(-t) = w(t)$ for all $t \in \mathcal{W}$.

Similarly, a discrete measure $\mathrm{d} \lambda_N (t) = \sum_{k=1}^N w_k \delta(t-t_k)$ is symmetric if $t_k = - t_{N+1-k}$, and $w_k = w_{N+1-k}$ for $k=1, 2, \dots, N$.

Theorem 1.17 states that: If $\mathrm{d} \lambda$ is symmetric, then

\[\pi_k(-t; \mathrm{d} \lambda) = (-1)^k \pi_k(t; \mathrm{d} \lambda), \quad k=0,1, \dots,\]

hence the parity of $k$ decides whether $\pi_k$ is even/odd.

Note

Symmetry is exploited in computeSP, where symmetry need not be relative to the origin, but some arbitrary point of the support.

Three-term Recurrence Relation

The fact that orthogonal polynomials can be represented in terms of a three-term recurrence formula is at the heart of all numerical methods of the package. The importance of the three-term recurrence relation is difficult to overestimate. It provides

  • efficient means of evaluating polynomials (and derivatives),
  • zeros of orthogonal polynomials by means of a eigenvalues of a symmetric, tridiagonal matrix
  • acces to quadrature rules,
  • normalization constants to create orthonormal polynomials.

Theorem 1.27 states:

Let $\pi_k(\cdot) = \pi_k(\cdot; \mathrm{d}\lambda)$ for $k = 0, 1, \dots$ be the monic orthogonal polynomials with respect to the measure $\mathrm{d} \lambda$. Then

\[\begin{aligned} \pi_{k+1}(t) &= (t - \alpha_k) \pi_k(t) - \beta_k \pi_{k-1}(t), \quad k= 0, 1, \dots, \\ \pi_o(t) &= 1, \\ \pi_{-1}(t) &= 0, \end{aligned}\]

where

\[\begin{aligned} \alpha = \alpha_k(\mathrm{d} \lambda) &= \frac{\langle t \pi_k, \pi_k \rangle_{\mathrm{d} \lambda}}{\langle \pi_k, \pi_k \rangle_{\mathrm{d} \lambda}}, &&k=0,1,2, \dots \\ \beta = \beta_k(\mathrm{d} \lambda) &= \frac{\langle \pi_k, \pi_k \rangle_{\mathrm{d} \lambda}}{\langle \pi_{k-1}, \pi_{k-1} \rangle_{\mathrm{d} \lambda}}, &&k=1,2,\dots \end{aligned}\]

Let $\tilde{\pi}_k(\cdot) = \tilde{\pi}_k(\cdot; \mathrm{d} \lambda t)$ denote the orthonormal polynomials, then

\[\begin{aligned} \sqrt{\beta_{k+1}} \tilde{\pi}_k(t) &= (t - \alpha_k) \tilde{\pi}_{k}(t) - \sqrt{\beta_k} \tilde{\pi}_{k-1}(t), \quad k = 0, 1, \dots \\ \tilde{\pi}_0(t) &= 1 \\ \tilde{\pi}_{-1}(t) &= 0. \end{aligned}\]

Note

Within the package, the coefficients (α,β) are the building block to represent (monic) orthogonal polynomials.

Notice that $\beta_0$ is arbitrary. Nevertheless, it is convenient to define it as

\[\beta_0(\mathrm{d}\lambda) = \langle \pi_0, \pi_0 \rangle_{\mathrm{d} \lambda} = \int_{\mathcal{W}} \mathrm{d} \lambda (t),\]

because it allows to compute the norms of the polynomials based on $\beta_k$ alone

\[\| \pi_n \|_{\mathrm{d} \lambda} = \beta_n(\mathrm{d} \lambda) \beta_{n-1}(\mathrm{d} \lambda) \cdots \beta_0(\mathrm{d} \lambda), \quad n = 0,1, \dots\]

Let the support be $\mathcal{W} = [a,b]$ for $0 < a,b < \infty$, then

\[\begin{aligned} & a < \alpha_k(\mathrm{d} \lambda) < b && k = 0,1,2, \dots \\ & 0 < \beta_k(\mathrm{d} \lambda) < \max(a^2, b^2) && k = 1, 2, \dots \end{aligned}\]

Quadrature Rules

An $n$-point quadrature rule for the measure $\mathrm{d} \lambda t$ is a formula of the form

\[\int_{\mathcal{W}} f(t) \mathrm{d} \lambda(t) = \sum_{\nu = 1}^{n} w_\nu f(\tau_\nu) + R_n(f).\]

The quadrature rule $\{ (\tau_\nu, w_\nu) \}_{\nu=1}^n$ composed of (mutually distinct) nodes $\tau_\nu$ and weights $w_\nu$ provides an approximation to the integral. The respective error is given by $R_n(f)$. If, for polynomials $p \in \mathcal{P}_d$, the error $R_n(p)$ vanishes, the respective quadrature rule is said to have a degree of exactness $d$. Gauss quadrature rule are special quadrature rules that have a degree of exactness $d = 2n - 1$. That means, taking a $n =3$-point quadrature rule, polynomials up to degree 5 can be integrated exactly. The nodes and weights for the Gauss quadrature rules have some remarkable properties:

  • all Gauss nodes are mutually distinct and contained in the interior of the support of $\mathrm{d} \lambda$;
  • the $n$ Gauss nodes are the zeros of $\pi_n$, the monic orthogonal polynomial of degree $n$ relative to the measure $\mathrm{d} \lambda$;
  • all Gauss weights are positive.

The Gauss nodes and weights can be computed using the Golub-Welsch algorithm. This means to solve an eigenvalue problem of a symmetric tridiagonal matrix.