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Orthogonal Polynomials — Definition, Formula & Examples

Orthogonal polynomials are a sequence of polynomials where each pair is orthogonal to every other pair with respect to a weighted inner product — meaning their inner product equals zero. Common families include Legendre, Hermite, and Chebyshev polynomials.

A sequence of polynomials {Pn(x)}n=0∞\{P_n(x)\}_{n=0}^{\infty}, where PnP_n has degree nn, is orthogonal with respect to a weight function w(x)≥0w(x) \geq 0 on an interval [a,b][a, b] if ∫abPm(x) Pn(x) w(x) dx=0\int_a^b P_m(x)\, P_n(x)\, w(x)\, dx = 0 whenever m≠nm \neq n.

Key Formula

∫abPm(x) Pn(x) w(x) dx=0for m≠n\int_a^b P_m(x)\, P_n(x)\, w(x)\, dx = 0 \quad \text{for } m \neq n
Where:
  • Pm,PnP_m, P_n = Polynomials of degree m and n in the orthogonal sequence
  • w(x)w(x) = Weight function, non-negative on [a, b]
  • [a,b][a, b] = Interval of integration (may be infinite)

How It Works

Orthogonality here extends the idea of perpendicular vectors to function spaces. Two polynomials PmP_m and PnP_n are orthogonal when their weighted integral over a specified interval is zero. The weight function w(x)w(x) and the interval [a,b][a,b] together define the inner product and determine which family of orthogonal polynomials you get. For instance, Legendre polynomials use w(x)=1w(x) = 1 on [−1,1][-1, 1], while Hermite polynomials use w(x)=e−x2w(x) = e^{-x^2} on (−∞,∞)(-\infty, \infty). You can construct orthogonal polynomials from any basis using the Gram-Schmidt process applied to {1,x,x2,…}\{1, x, x^2, \ldots\}.

Worked Example

Problem: Verify that the Legendre polynomials P0(x)=1P_0(x) = 1 and P1(x)=xP_1(x) = x are orthogonal on [−1,1][-1, 1] with weight w(x)=1w(x) = 1.
Set up the inner product: Compute the integral of the product of the two polynomials over the interval.
∫−11P0(x) P1(x) dx=∫−111⋅x dx\int_{-1}^{1} P_0(x)\, P_1(x)\, dx = \int_{-1}^{1} 1 \cdot x\, dx
Evaluate the integral: The antiderivative of xx is x22\frac{x^2}{2}. Evaluate at the bounds.
[x22]−11=12−12=0\left[\frac{x^2}{2}\right]_{-1}^{1} = \frac{1}{2} - \frac{1}{2} = 0
Answer: The integral equals 00, confirming that P0(x)P_0(x) and P1(x)P_1(x) are orthogonal on [−1,1][-1, 1].

Why It Matters

Orthogonal polynomials are essential in numerical analysis for Gaussian quadrature (highly accurate numerical integration) and in approximation theory for expanding functions into series with minimal error. They also appear in quantum mechanics, signal processing, and solving differential equations via spectral methods.

Common Mistakes

Mistake: Forgetting the weight function and testing orthogonality with w(x)=1w(x) = 1 for all families.
Correction: Each family of orthogonal polynomials has its own specific weight function. Hermite polynomials, for example, are only orthogonal under w(x)=e−x2w(x) = e^{-x^2}, not under w(x)=1w(x) = 1.