Orthogonal functions

In mathematics, orthogonal functions belong to a function space which is a vector space (usually over R) 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: $\langle f,g\rangle =\int f(x)g(x)\,dx$

Then functions f and g are orthogonal when this integral is zero: $\langle f,\ g\rangle =0.$ As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space.

Trigonometric functions

Main article: Fourier series

Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions, sin nx, are orthogonal on the interval (-π, π), if m ≠ n. For then

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

so that the integral of the product of the two sines 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.

When the function space consists of polynomials, the orthogonal functions are orthogonal polynomials, which have several varieties.

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.

Other sets of orthogonal functions

• Bessel functions
• Hermite polynomials
• Laguerre polynomials
• Legendre polynomials
• Spherical harmonics
• Walsh functions
• Zernike polynomials
• Chebyshev polynomials

See also

• Hilbert space
• Harmonic analysis
• Orthonormal basis
• Eigenfunction
• Eigenvalues and eigenvectors

References

• George B. Arfken & Hans J. Weber (2005) Mathematical Methods for Physicists, 6th edition, chapter 10: Sturm-Liouville Theory — Orthogonal Functions, Academic Press.

External links

• Orthogonal Functions, on MathWorld.