Orthogonal Functions

Orthogonal Functions

In mathematics, two functions and are called orthogonal if their inner product 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

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

For an intuitive perspective on this inner product, suppose approximating vectors and 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 and, 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:

• Sines and cosines
• Hermite polynomials
• Legendre polynomials
• Spherical harmonics
• Walsh functions
• Zernike polynomials
• Chebyshev polynomials