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.

Choice of inner product[edit]

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).[1]

In differential equations[edit]

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[edit]

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.

See also[edit]