Inmathematics,orthogonal functions belong to afunction space that is avector space equipped with abilinear form. When the function space has aninterval as thedomain, the bilinear form may be theintegral of the product of functions over the interval:

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

The functions${\displaystyle f}$ and${\displaystyle g}$ areorthogonal when this integral is zero, i.e.${\displaystyle ⟨f,g⟩=0}$ whenever${\displaystyle f\ne g}$. As with abasis 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 vectordot product; two vectors are mutually independent (orthogonal) if their dot-product is zero.

Suppose${\displaystyle \{f_0,f_1,\dots \}}$ is a sequence of orthogonal functions of nonzeroL²-norms${\Vert f_n\Vert }_2=\sqrt{⟨f_n,f_n⟩}={\left(\int f_n^{2}dx\right)}^{\frac{1}{2}}$. It follows that the sequence${\displaystyle \left\{f_n/{\Vert f_n\Vert }_2\right\}}$ is of functions ofL²-norm one, forming anorthonormal sequence. To have a definedL²-norm, the integral must be bounded, which restricts the functions to beingsquare-integrable.

Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functionssinnx andsinmx are orthogonal on the interval${\displaystyle x\in (-\pi ,\pi )}$ when${\displaystyle m\ne n}$ andn andm 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 atrigonometric polynomial to approximate a given function on the interval with itsFourier series.

If one begins with themonomial sequence${\displaystyle \left\{1,x,x^2,\dots \right\}}$ on the interval${\displaystyle [-1,1]}$ and applies theGram–Schmidt process, then one obtains theLegendre polynomials. Another collection of orthogonal polynomials are theassociated Legendre polynomials.

The study of orthogonal polynomials involvesweight functions${\displaystyle w(x)}$ that are inserted in the bilinear form:

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

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

Both physicists and probability theorists useHermite polynomials on${\displaystyle (-\mathrm{\infty },\mathrm{\infty })}$, 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$w(x)=\frac{1}{\sqrt{1-x^2}}$ or$w(x)=\sqrt{1-x^2}$.

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

Walsh functions andHaar wavelets are examples of orthogonal functions with discrete ranges.

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 theCayley transform first, to bring the argument into[−1, 1]. This procedure results in families ofrational orthogonal functions calledLegendre rational functions andChebyshev rational functions.

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

• Eigenvalues and eigenvectors
• Hilbert space
• Karhunen–Loève theorem
• Lauricella's theorem
• Wannier function

• George B. Arfken & Hans J. Weber (2005)Mathematical Methods for Physicists, 6th edition, chapter 10: Sturm-Liouville Theory — Orthogonal Functions,Academic Press.
• Price, Justin J. (1975)."Topics in orthogonal functions".American Mathematical Monthly.82:594–609.doi:10.2307/2319690. Archived fromthe original on 2021-01-15. Retrieved2019-02-09.
• Giovanni Sansone (translated by Ainsley H. Diamond) (1959)Orthogonal Functions,Interscience Publishers.

• Orthogonal Functions, on MathWorld.

• Functional analysis
• Types of functions

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• Short description matches Wikidata