Orthogonal and Orthonormal Functions: Definition, Examples

The orthogonal and orthonormal functions on an interval [a, b] are such functions where the tangents to the curves y=Φ₁(x) and y=Φ₂(x) at their intersecting points are perpendicular to each other. In this article, we study orthogonal and orthonormal functions along with examples.

Orthogonal Function Definition

A set of functions {Φ₁(x), Φ₂(x), …, Φ_n(x), …} is called orthogonal on the interval [a, b] if

$\int_a^b \phi_i(x) \phi_j(x)~dx=0$ ∀ i ≠ j.

For example, {sinx, cosx} are orthogonal functions on [-π, π].

Geometrical Interpretation:

Two functions Φ₁(x) and Φ₂(x) are orthogonal in the interval [a, b] means that the tangent to the curves y=Φ₁(x) and y=Φ₂(x) at their intersecting points are orthogonal, that is, perpendicular to each other.

Orthogonal Function Example

Ex1: The functions
Φ₁(x) = sinx and Φ₂(x) = cosx on [-π, π] are orthogonal.

Proof: To show they are orthogonal functions, we compute

$\int_{-\pi}^\pi \phi_1(x) \phi_2(x)~dx$

= $\int_{-\pi}^\pi \sin x \cos x~dx$

= $\dfrac{1}{2} \int_{-\pi}^\pi \sin 2x~dx$ using the trigonometric identity sin2x = 2sinx cosx.

= 0 as sin2x is an odd function.

Therefore, sinx and cosx are orthogonal functions on [-π, π].

Related Articles: Even and Odd Functions

Norm of Function

The norm of a function Φ(x), denoted by ||Φ(x)||, is defined as follows:

$\boxed{||\phi(x)|| = \left[ \int_a^b \phi^2(x)~dx\right]^{1/2}}$

Example:

The norm of sinx on [-π, π] is equal to √π which can be calculated as follows.

Norm of sinx = ||sinx||

= $\left[ \int_{-\pi}^\pi \sin^2x~dx\right]^{1/2}$

= $\left[ \dfrac{1}{2} \int_{-\pi}^\pi 2\sin^2x~dx\right]^{1/2}$

= $\left[ \dfrac{1}{2} \int_{-\pi}^\pi (1-\cos 2x)~dx\right]^{1/2}$ as we know 2sin²x = 1-cos2x.

= $\left[ \dfrac{1}{2} \cdot 2 \int_{0}^\pi (1-\cos 2x)~dx\right]^{1/2}$ as 1-cos2x is an even function.

= $\left\{\left[ x-\dfrac{\sin 2x}{2}\right]_{0}^\pi \right\}^{1/2}$

= √π.

Therefore, the norm of sin x on [-π, π] is equal to √π.

Similarly, the norm of cosx on [-π, π] is equal to √π.

Orthonormal Function Definition

The functions Φ₁(x), Φ₂(x), … are said to be orthonormal functions on [a, b] if the functions Φ_i(x) are orthogonal with norm 1. That is,

1. $\int_a^b \phi_i(x) \phi_j(x)~dx=0$ ∀ i ≠ j.
2. $\int_a^b \phi_i^2(x) ~dx=0$ ∀ i.

Orthonormal Function Example

Let us consider the set of functions {1, cosx, sinx, cos2x, sin2x, …} on the interval [-π, π]. Lets now check its orthogonality.

As cosmx is an even function and sinmx is an odd function, for all m = 1, 2, 3, … we deduce the following:

$\int_{-\pi}^\pi 1 \cdot \cos mx ~ dx$ $=2\int_0^\pi \cos mx dx$ $=2\left[ \dfrac{\sin mx}{m} \right]_0^\pi$ = 0. $\int_{-\pi}^\pi 1 \cdot \sin mx ~ dx$ $=\int_{-\pi}^\pi \sin mx dx$ = 0.

Also,

$\int_{-\pi}^\pi \sin mx \cdot \sin nx ~ dx$ = $\dfrac{1}{2} \int_{-\pi}^\pi [\cos(m-n)x – \cos (m+n)x] ~ dx$ = $\dfrac{1}{2} \times 2$ $\int_{0}^\pi [\cos(m-n)x – \cos (m+n)x] ~ dx$ = $\left[\dfrac{\sin(m-n)x}{m-n} – \dfrac{\sin (m+n)x}{m+n} \right]_{0}^\pi$ = 0 for all m, n with m≠n.

And,

$\int_{-\pi}^\pi \sin mx \cdot \cos nx ~ dx$ = $\dfrac{1}{2} \int_{-\pi}^\pi [\sin(m+n)x + \sin (m-n)x] ~ dx$ = 0 for all m, n with m≠n. Similarly, $\int_{-\pi}^\pi \cos mx \cdot \cos nx ~ dx$ = $\dfrac{1}{2} \int_{-\pi}^\pi [\cos(m+n)x + \cos (m-n)x] ~ dx$ = 0 for all m, n with m≠n.

Therefore, the set {1, cosx, sinx, cos2x, sin2x, …} is an orthogonal set.

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