Orthogonal Polynomial Sequence, Definition, Solved Examples

An orthogonal polynomial sequence is a family of sequences in which any two distinct polynomials are orthogonal with respect to an inner product. In this page, we will study about orthogonal polynomial sequence with its definition and examples.

Orthogonal Polynomial Sequence

Definition 1: A sequence of polynomials {P_n(x)} is called orthogonal over the interval (a,b) if

$\displaystyle \int_a^b P_m(x)P_n(x) dx=0$ for m≠n.

Definition 2: A sequence of polynomials {P_n(x)} is called orthogonal with respect to the weight function w(x) over the interval (a,b) if

$\displaystyle \int_a^b P_m(x)P_n(x) w(x) dx=0$ for m≠n, where w(x)>0 ∀ x.

Solved Examples

Ex1: Prove that the sequence $\{\cos n\theta\}_{n=1}^\infty$ is an orthogonal sequence over the interval $(0, \pi)$.

Answer:

Here P_n(θ) = cos(nθ).

For m≠n, we have:

$\displaystyle \int_0^\pi P_m(\theta)P_n(\theta) \ d\theta$

= $\displaystyle \int_0^\pi \cos(m\theta) \cos(n\theta) \ d\theta$

= $\dfrac{1}{2} \displaystyle \int_0^\pi 2\cos(m\theta) \cos(n\theta) \ d\theta$

= $\dfrac{1}{2} \displaystyle \int_0^\pi [\cos(m+n)\theta + \cos(m-n)\theta] \ d\theta$ using the formula 2cosa cosb = cos(a+b) + cos(a-b).

= $\dfrac{1}{2} \left[ \dfrac{\sin(m+n)\theta}{m+n} + \dfrac{\sin(m-n)\theta}{m-n} \right]_0^\pi$

= 0 as we know that sin(nπ) = 0.

This shows that the sequence $\{\cos n\theta\}_{n=1}^\infty$ is an orthogonal sequence over the interval $(0, \pi)$.

Ex2: Show that the sequence $\{T_n(x)\}_{n=1}^\infty$ where T_n(x) = cos nθ = cos(n cos^-1x), -1 ≤ x ≤ 1, is an orthogonal polynomial sequence over (0, 1) with respect to the weight $(1-x^2)^{-1/2}$.

Answer:

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