What is "orthogonal functions"

n. (orthogonal function English)

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:

⟨f, g⟩ = ∫f(x)g(x) dx.

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

Suppose {f}, n = 0, 1, 2, … is a sequence of orthogonal functions. If f has positive support then ⟨f, f⟩ = ∫fdx = m is the L2-norm of f, and the sequence $\{ \frac {f_n}{m_n} \}$ has functions of L2-norm one, forming an orthonormal sequence. The possibility that an integral is unbounded must be avoided, hence attention is restricted to square-integrable functions.