Orthogonal functions

In mathematics, orthogonal functions belong to a function space that is a vector space equipped with 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:

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

The functions ${\displaystyle f}$ and ${\displaystyle g}$ are orthogonal when this integral is zero, i.e. ${\displaystyle \langle f,\,g\rangle =0}$ whenever ${\displaystyle f\neq g}$. As with a basis 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 vector dot product; two vectors are mutually independent (orthogonal) if their dot-product is zero.

Suppose ${\displaystyle \{f_{0},f_{1},\ldots \}}$ is a sequence of orthogonal functions of nonzero L²-norms ${\textstyle \left\|f_{n}\right\|_{2}={\sqrt {\langle f_{n},f_{n}\rangle }}=\left(\int f_{n}^{2}\ dx\right)^{\frac {1}{2}}}$. It follows that the sequence ${\displaystyle \left\{f_{n}/\left\|f_{n}\right\|_{2}\right\}}$ is of functions of L²-norm one, forming an orthonormal sequence. To have a defined L²-norm, the integral must be bounded, which restricts the functions to being square-integrable.