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 the 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).

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