Orthogonal functions

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In mathematics, two functions $f$ and $g$ are called orthogonal if their inner product $\langle f,g\rangle$ is zero for f ≠ g. How the inner product of two functions is defined may vary depending on context. However, a typical definition of an inner product for functions is

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

with appropriate integration boundaries. Here, the asterisk indicates the complex conjugate of f.

For another perspective on this inner product, suppose approximating vectors ${\vec {f}}$ and ${\vec {g}}$ 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 ${\vec {f}}$ and ${\vec {g}}$ , 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).

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:

Generalization of vectors

It can be shown that orthogonality of functions is a generalization of the concept of orthogonality of vectors. Suppose we define V to be the set of variables on which the functions f and g operate. (In the example above, V={x} since x is the only parameter to f and g. Since there is one parameter, one integral sign is required to determine orthogonality. If V contained two variables, it would be necessary to integrate twice—over a range of each variable—to establish orthogonality.) If V is an empty set, then f and g are just constant vectors, and there are no variables over which to integrate. Thus, the equation reduces to a simple inner-product of the two vectors.

Orthogonal functions

Generalization of vectors

See also