Orthogonal functions

In mathematics, orthogonal functions belong to a function space that is a vector space equipped with 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 and are orthogonal when this integral is zero, i.e. whenever . As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot product; two vectors are mutually independent (orthogonal) if their dot-product is zero. Suppose is a sequence of orthogonal functions of nonzero L2-norms . It follows that the sequence is of functions of L2-norm one, forming an orthonormal sequence. To have a defined L2-norm, the integral must be bounded, which restricts the functions to being square-integrable. Fourier series and Harmonic analysis Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions sin nx and sin mx are orthogonal on the interval when and n and m are positive integers. For then and the integral of the product of the two sine functions vanishes. Together with cosine functions, these orthogonal functions may be assembled into a trigonometric polynomial to approximate a given function on the interval with its Fourier series. Orthogonal polynomials If one begins with the monomial sequence on the interval and applies the Gram–Schmidt process, then one obtains the Legendre polynomials. Another collection of orthogonal polynomials are the associated Legendre polynomials. The study of orthogonal polynomials involves weight functions that are inserted in the bilinear form: For Laguerre polynomials on the weight function is . Both physicists and probability theorists use Hermite polynomials on , where the weight function is or . Chebyshev polynomials are defined on and use weights or . Zernike polynomials are defined on the unit disk and have orthogonality of both radial and angular parts.