# Orthogonal functions

In mathematics, **orthogonal functions** belong to a function space which is a vector space (usually over R) that has a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:

Then functions *f* and *g* are orthogonal when this integral is zero:
As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space.

Suppose {*f*_{n}}, n = 0, 1, 2, … is a sequence of orthogonal functions. If *f*_{n} has positive support then is the L2-norm of *f*_{n}, and the sequence has functions of L2-norm one, forming an orthonormal sequence. The possibility that an integral is unbounded must be avoided, hence attention is restricted to square-integrable functions.

## Trigonometric functions

Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions, sin *nx* and sin *mx*, are orthogonal on the interval (-π, π), if *m* ≠ *n*. For then

so that the integral of the product of the two sines vanishes.^{[1]} Together with cosine functions, these orthogonal functions may be assembled into a trigonometric polynomial to approximate a given function on the interval with its Fourier series.

## Polynomials

If one begins with the monomial sequence {1, *x*, *x*^{2}, ... *x*^{n} ...} on [–1, 1] and applies the Gram-Schmidt process, then one obtains the Legendre polynomials. Another collection of orthogonal polynomials are the associated Legendre polynomials.

The study of orthogonal polynomials involves weight functions *w*(*x*) which are inserted in the bilinear form:

For Laguerre polynomials on (0, ∞) the weight function is

Both physicists and probability theorists use Hermite polynomials on (−∞, ∞) where the weight function is or

Chebyshev polynomials are defined on [−1, 1] and use weights or

Zernike polynomials are defined on the unit disk and have orthogonality of both radial and angular parts.

## Binary-valued functions

Walsh functions and Haar wavelets are examples of orthogonal functions with discrete ranges.

## Rational functions

Legendre and Chebyshev polynomials provide orthogonal families for the interval [−1, 1] while occasionally orthogonal families are required on [0, ∞). In this case it is convenient to apply the Cayley transform first, to bring the argument into [−1, 1]. This procedure results in families of rational orthogonal functions called Legendre rational functions and Chebyshev rational functions.

## In differential equations

Solutions of linear differential equations with boundary conditions can often be written as a weighted sum of orthogonal solution functions (a.k.a. eigenfunctions), leading to generalized Fourier series.

## See also

- Hilbert space
- Eigenvalues and eigenvectors
- Wannier function
- Lauricella's theorem
- Karhunen-Loeve theorem

## References

- ↑ Antoni Zygmund (1935)
*Trigonometrical Series*, page 6, Mathematical Seminar, University of Warsaw

- George B. Arfken & Hans J. Weber (2005)
*Mathematical Methods for Physicists*, 6th edition, chapter 10: Sturm-Liouville Theory — Orthogonal Functions, Academic Press. - Giovanni Sansone (translated by Ainsley H. Diamond) (1959)
*Orthogonal Functions*, Interscience Publishers.

## External links

- Orthogonal Functions, on MathWorld.