Orthogonal functions

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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:

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

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.

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

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

and the integral of the product of the two sine functions vanishes.[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 ${\displaystyle \{1,x,x^{2},\dots \}}$ 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 ${\displaystyle w(x)=e^{-{\frac {x^{2}}{2}}}.}$

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

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.