Orthogonal Polynomials

Orthogonality for Functions

Use real continuous functions on a finite interval $a<b$ and an integrable weight that is positive in the interval interior. Integrable endpoint singularities are allowed, as in the Chebyshev weight below.

The weighted inner product and its induced norm are

\langle f,g\rangle_w=\int_a^b f(x)g(x)w(x)\,dx

Two functions are orthogonal when $\langle f,g\rangle_w=0$ . This weighted $L^2$ geometry measures the weighted squared error across the interval.

\begin{aligned} &\|f-g\|_w^2 \\ &=\int_a^b|f(x)-g(x)|^2w(x)\,dx \end{aligned}

By contrast, the maximum norm measures the $largest pointwise error$ .

\|f-g\|_\infty=\max_{x\in[a,b]}|f(x)-g(x)|

Orthogonal projection minimizes the weighted $L^2$ error. Chebyshev polynomials also have a separate minimax property. The weighted inner product and the maximum norm measure different errors.

Projection onto an Orthogonal Basis

Suppose $p_0,\ldots,p_n$ are mutually orthogonal. The degree- $n$ projection of $f$ is

\Pi_n f(x)=\sum_{k=0}^n c_kp_k(x)

Because cross terms vanish, $each coefficient can be computed independently$ . This is the functional analogue of resolving a vector along perpendicular coordinate axes.

Building the Sequence Step by Step

Applying Gram-Schmidt to $1,x,x^2,\ldots$ produces a monic orthogonal sequence $\pi_k$ . Under a positive inner product it satisfies

\pi_0(x)=1

The recurrence coefficients are

\alpha_k=\frac{\langle x\pi_k,\pi_k\rangle_w}{\|\pi_k\|_w^2}

The three-term recurrence and $\beta_k$ apply for $k\ge1$ .

Multiplication by $x$ raises the degree by one. Orthogonality removes every component below $\pi_{k-1}$ , so the recurrence needs only $three terms$ .

Chebyshev Polynomials of the First Kind

On $[-1,1]$ , choose

w(x)=\frac{1}{\sqrt{1-x^2}}

T_0(x)=1,\qquad T_1(x)=x

Their weighted orthogonality is

\begin{aligned} &\int_{-1}^{1}\frac{T_j(x)T_k(x)}{\sqrt{1-x^2}}\,dx \\ &=\begin{cases}\pi,&j=k=0,\\\frac{\pi}{2},&j=k>0,\\0,&j\ne k.\end{cases} \end{aligned}

Since $|T_k(x)|\le1$ on $[-1,1]$ and the leading coefficient is $2^{k-1}$ for $k\ge1$ , the monic polynomial $2^{1-k}T_k$ has the $smallest maximum norm$ among all monic degree- $k$ polynomials. It therefore solves the minimax approximation problem for monic polynomials of degree $k$ .

The first Chebyshev polynomials

T_0,T_1,T_2,T_3

All four remain between

-1

and

1

on the interval

[-1,1]

$x$: $[-1, 1]$
$y$: $[-1, 1]$

Legendre Polynomials

For the unweighted inner product on $[-1,1]$ ,

\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)\,dx

For this inner product, the standard Legendre polynomials satisfy

P_0(x)=1,\qquad P_1(x)=x

Their orthogonality and normalization are

\int_{-1}^{1}P_j(x)P_k(x)\,dx=\frac{2}{2k+1}\delta_{jk}

For example,

P_2(x)=\frac12(3x^2-1)

The second identity gives the $exact orthogonal expansion$ .

Changing the Interval of Orthogonal Polynomials

To map the interval $[-1,1]$ to $[a,b]$ , use the following $affine change of variables$ .

x=\frac{a+b}{2}+\frac{b-a}{2}t

If $\varphi_k$ is orthonormal on $[-1,1]$ , then

\widetilde{\varphi}_k(x)=\sqrt{\frac{2}{b-a}}\,\varphi_k\left(\frac{2x-a-b}{b-a}\right)

The function $\widetilde{\varphi}_k$ is orthonormal on $[a,b]$ because

\begin{aligned} &\int_a^b\widetilde{\varphi}_j(x)\widetilde{\varphi}_k(x)\,dx \\ &=\int_{-1}^{1}\varphi_j(t)\varphi_k(t)\,dt \\ &=\delta_{jk} \end{aligned}

Connection to Gaussian Quadrature

Let $p_n$ be the degree- $n$ polynomial orthogonal for the same positive weight and interval as an integral. Its $n$ roots are the Gaussian quadrature nodes. Suitable positive weights $w_i$ then give

\int_a^b f(x)w(x)\,dx\approx\sum_{i=1}^n w_i f(x_i)

The rule is $exact for every polynomial$ of degree at most $2n-1$ . Legendre roots produce Gauss-Legendre quadrature for the weight $w(x)=1$ . Chebyshev roots produce Gauss-Chebyshev quadrature for $w(x)=1/\sqrt{1-x^2}$ .

For two Gauss-Legendre nodes, $P_2(x)=0$ gives $x=\pm1/\sqrt3$ and both weights are 1. Check the quadratic integral directly:

\int_{-1}^1x^2\,dx=\frac23=1\left(-\frac1{\sqrt3}\right)^2+1\left(\frac1{\sqrt3}\right)^2

Basis	Weight on $[-1,1]$	Best suited to
Chebyshev $T_k$	$1/\sqrt{1-x^2}$	stable interpolation grids, near-minimax approximation, spectral methods
Legendre $P_k$	$1$	unweighted $L^2$ projection and Gauss-Legendre quadrature

Always match the polynomial family, inner product, approximation objective, and quadrature weight. Changing one of them changes the problem being solved.