Orthogonal functions: Difference between revisions

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As with a [[basis (linear algebra)|basis]] of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space.

Suppose <math> \{ f_n \}, n = 0, 1, 2, \ldots</math> is a sequence of orthogonal functions. If <math>f_n</math> has positive [[support (mathematics)|support]] then <math>\langle f_n, f_n \rangle = \int f_n ^2 \ dx = m_n </math> is the [[L2-norm|''L''<subsup>2</subsup>-norm]] of <math>f_n</math>, and the sequence <math>\left\{ \frac {f_n}{m_n} \right\}</math> has functions of ''L''<subsup>2</subsup>-norm one, forming an [[orthonormal sequence]]. The possibility that an integral is unbounded must be avoided, hence attention is restricted to [[square-integrable function]]s.

==Trigonometric functions==