Orthogonal functions

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:

The functions f and g are orthogonal when this integral is zero: ${\displaystyle \langle f,\ g\rangle =0.}$ 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}, n = 0, 1, 2, … is a sequence of orthogonal functions. If f_n has positive support then ${\displaystyle \langle f_{n},f_{n}\rangle =\int f_{n}^{2}\ dx=m_{n}}$ is the L2-norm of f_n, and the sequence ${\displaystyle \{{\frac {f_{n}}{m_{n}}}\}}$ 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.

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