# 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:
    $\langle f,g\rangle =\int {\overline {f(x)}}g(x)\,dx.$
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 L$^2$-norms ‖f$_n$‖$_2$=√(⟨f$_n$, f$_n$⟩)=(∫f$_n$$^2$ dx)$^{1/2}${\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 $\left\{f_{n}/\left\|f_{n}\right\|_{2}\right\}$ is of functions of L$^2$-norm one, forming an orthonormal sequence. To have a defined L$^2$-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
    $2\sin \left(mx\right)\sin \left(nx\right)=\cos \left(\left(m-n\right)x\right)-\cos \left(\left(m+n\right)x\right),$
and the integral of the product of the two sine functions vanishes. 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 $\left\{1,x,x^{2},\dots \right\}$ 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 $w(x)=e^{-x^{2}/2}$.
Chebyshev polynomials are defined on $[-1,1]$ and use weights w(x)=1/√(1−x$^2$) {\textstyle w(x)={\frac {1}{\sqrt {1-x^{2}}}}} or w(x)=√(1−x$^2$){\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

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.