Orthogonal functions

In mathematics, orthogonal functions belong to a function space which is a vector space 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: In mathematics, orthogonal functions belong to a function space which is a vector space 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 {displaystyle f} and g {displaystyle g} are orthogonal when this integral is zero, i.e. ⟨ f ,   g ⟩ = 0 {displaystyle langle f, g angle =0} whenever f ≠ g {displaystyle f eq g} . As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Suppose { f 0 , f 1 , … } {displaystyle {f_{0},f_{1},ldots }} is a sequence of orthogonal functions of nonzero L2-norms ‖ f n ‖ 2 = ⟨ f n , f n ⟩ = ( ∫ f n 2   d x ) 1 2 {displaystyle Vert f_{n}Vert _{2}={sqrt {langle f_{n},f_{n} angle }}=left(int f_{n}^{2} dx ight)^{frac {1}{2}}} . It follows that the sequence { f n / ‖ f n ‖ 2 } {displaystyle left{f_{n}/Vert f_{n}Vert _{2} ight}} 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. 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 x ∈ ( − π , π ) {displaystyle xin (-pi ,pi )} when m ≠ n {displaystyle m eq n} 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. If one begins with the monomial sequence { 1 , x , x 2 , … } {displaystyle {1,x,x^{2},dots }} on the interval [ − 1 , 1 ] {displaystyle } 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 w ( x ) {displaystyle w(x)} that are inserted in the bilinear form: For Laguerre polynomials on ( 0 , ∞ ) {displaystyle (0,infty )} the weight function is w ( x ) = e − x {displaystyle w(x)=e^{-x}} . Both physicists and probability theorists use Hermite polynomials on ( − ∞ , ∞ ) {displaystyle (-infty ,infty )} , where the weight function is w ( x ) = e − x 2 {displaystyle w(x)=e^{-x^{2}}} or w ( x ) = e − x 2 2 . {displaystyle w(x)=e^{-{frac {x^{2}}{2}}}.}