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Orthogonal Polynomials: Simple Definition, Examples

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Orthogonal polynomials (also called an orthogonal polynomial sequence) are a set of polynomials that are orthogonal (perpendicular, or at right angles) to each other.

As a simple example, the two-dimensional coordinates {x, y} are perpendicular to each other. So two polynomials that each fit along the x and y axes are orthogonal to each other. When we talk about “orthogonal polynomials” though, we actually mean an orthogonal polynomial sequence. In other words, there must be an infinite number of them in order to meet the formal definition.

Formal Definition

Orthogonal polynomials are the infinite sequence:

p0 (x), p1 (x), p2 (x), … pn (x),

Where:

  • pn(x) is a polynomial with degree n,
  • Any two polynomials are orthogonal to each other.

This can be represented by the following integral, which basically means if you multiply the two functions and integrate the result is zero:


The closed interval [a, b] is called the interval of orthogonality; The interval can be infinite at one end, or both.

As an example, of what this integral means, the following image shows the separate integration of two orthogonal polynomials ½(3x2 – 1) and ½(5x3 – 3x) on the closed interval [-1, 1]:
orthogonal polynomials integrals

It should come as no surprise then, that the two integrals, when multiplied together on the same interval (see: integration by parts), also equal zero.


Examples of Orthogonal Polynomials

The above two integrals (graphed with Integral-Calculator) are a part of a sequence called Legendre polynomials, which form solutions to the Legendre differential equation. Another set of orthogonal polynomials which are widely used are Hermite polynomials, which are part of the solution to the quantum harmonic oscillator Hamiltonian.

References

Arfken, G. “Orthogonal Polynomials.” Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 520-521, 1985.
Iyanaga, S. and Kawada, Y. (Eds.). “Systems of Orthogonal Functions.” Appendix A, Table 20 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1477, 1980
Sansone, G. Orthogonal Functions. New York: Dover, 1991
Orthogonal Polynomials. Retrieved February 12, 2020 from: https://sydney.edu.au/science/chemistry/~mjtj/CHEM3117/Resources/poly_etc.pdf

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Stephanie Glen. "Orthogonal Polynomials: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/orthogonal-polynomials/
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