Orthogonal functions

In mathematics, two functions $f$ and $g$ are called orthogonal if their inner product $\langle f,g\rangle$ is zero for f ≠ g.

Choice of inner product

How the inner product of two functions is defined may vary depending on context. However, a typical definition of an inner product for functions is

$\langle f,g\rangle = \int f(x) ^* g(x)\,dx$

with appropriate integration boundaries. Here, the asterisk indicates the complex conjugate of f.

For another perspective on this inner product, suppose approximating vectors $\vec{f}$ and $\vec{g}$ are created whose entries are the values of the functions f and g, sampled at equally spaced points. Then this inner product between f and g can be roughly understood as the dot product between approximating vectors $\vec{f}$ and $\vec{g}$, in the limit as the number of sampling points goes to infinity. Thus, roughly, two functions are orthogonal if their approximating vectors are perpendicular (under this common inner product).[1]

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.

Examples

Examples of sets of orthogonal functions:

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

See also

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