# Orthogonal functions

(Redirected from Orthogonal function)

In mathematics, orthogonal functions belong to a function space which is a vector space 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:

$\langle f,g\rangle =\int {\overline {f(x)}}g(x)\,dx.$ The functions $f$ and $g$ are orthogonal when this integral is zero, i.e. $\langle f,\ g\rangle =0$ whenever $f\neq g$ . As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot-product; two vectors are mutually independent (orthogonal) if their dot-product is zero.

Suppose $\{f_{0},f_{1},\ldots \}$ is a sequence of orthogonal functions of nonzero L2-norms $\Vert f_{n}\Vert _{2}={\sqrt {\langle f_{n},f_{n}\rangle }}=\left(\int f_{n}^{2}\ dx\right)^{\frac {1}{2}}$ . It follows that the sequence $\left\{f_{n}/\Vert f_{n}\Vert _{2}\right\}$ is of functions of L2-norm one, forming an orthonormal sequence. To have a defined L2-norm, the integral must be bounded, which restricts the functions to being square-integrable.

## 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 $x\in (-\pi ,\pi )$  when $m\neq n$  and n and m are positive integers. For then

$2\sin(mx)\sin(nx)=\cos \left((m-n)x\right)-\cos \left((m+n)x\right),$

and the integral of the product of the two sine functions vanishes. 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 $\{1,x,x^{2},\dots \}$  on the interval $[-1,1]$  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 $w(x)$  that are inserted in the bilinear form:

$\langle f,g\rangle =\int w(x)f(x)g(x)\,dx.$

For Laguerre polynomials on $(0,\infty )$  the weight function is $w(x)=e^{-x}$ .

Both physicists and probability theorists use Hermite polynomials on $(-\infty ,\infty )$ , where the weight function is $w(x)=e^{-x^{2}}$  or $w(x)=e^{-{\frac {x^{2}}{2}}}.$

Chebyshev polynomials are defined on $[-1,1]$  and use weights $w(x)={\frac {1}{\sqrt {1-x^{2}}}}$  or $w(x)={\sqrt {1-x^{2}}}$ .

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.