Orthogonal functions

In mathematics, two functions ${\displaystyle f}$ and ${\displaystyle g}$ are called orthogonal if their inner product ${\displaystyle \langle f,g\rangle }$ is zero. Whether or not two particular functions are orthogonal depends on how their inner product has been defined. A typical definition of an inner product for functions is

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

with appropriate integration boundaries. Here, the star is the complex conjugate.

For an intuitive perspective on this inner product, suppose approximating vectors ${\displaystyle {\vec {f}}}$ and ${\displaystyle {\vec {g}}}$ 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 ${\displaystyle {\vec {f}}}$ and ${\displaystyle {\vec {g}}}$, 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]

See also Hilbert space for a more rigorous background.

Solutions of linear differential equations with boundary conditions can often be written as a weighted sum of orthogonal solution functions (a.k.a. eigenfunctions).

Examples of sets of orthogonal functions:

• Hermite polynomials
• Legendre polynomials
• Spherical harmonics
• Walsh functions

See also

• Orthogonal polynomials.