# Orthogonal functions

In mathematics, orthogonal functions belong to a function space that is a vector space equipped with 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:

${\displaystyle \langle f,g\rangle =\int {\overline {f(x)}}g(x)\,dx.}$

The functions ${\displaystyle f}$ and ${\displaystyle g}$ are orthogonal when this integral is zero, i.e. ${\displaystyle \langle f,\,g\rangle =0}$ whenever ${\displaystyle f\neq g}$. 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 ${\displaystyle \{f_{0},f_{1},\ldots \}}$ is a sequence of orthogonal functions of nonzero L2-norms ${\textstyle \left\|f_{n}\right\|_{2}={\sqrt {\langle f_{n},f_{n}\rangle }}=\left(\int f_{n}^{2}\ dx\right)^{\frac {1}{2}}}$. It follows that the sequence ${\displaystyle \left\{f_{n}/\left\|f_{n}\right\|_{2}\right\}}$ 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 ${\displaystyle x\in (-\pi ,\pi )}$ when ${\displaystyle m\neq n}$ and n and m are positive integers. For then

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

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

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

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

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

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

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

Plot of the Chebyshev rational functions of order n=0,1,2,3 and 4 between x=0.01 and 100.

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.