Orthogonal functions (original) (raw)

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Type of function

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:

⟨ f , g ⟩ = ∫ f ( x ) ¯ g ( x ) d x . {\displaystyle \langle f,g\rangle =\int {\overline {f(x)}}g(x)\,dx.} {\displaystyle \langle f,g\rangle =\int {\overline {f(x)}}g(x)\,dx.}

The functions f {\displaystyle f} {\displaystyle f} and g {\displaystyle g} {\displaystyle g} are orthogonal when this integral is zero, i.e. ⟨ f , g ⟩ = 0 {\displaystyle \langle f,\,g\rangle =0} {\displaystyle \langle f,\,g\rangle =0} whenever f ≠ g {\displaystyle f\neq g} {\displaystyle f\neq g}. 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 { f 0 , f 1 , … } {\displaystyle \{f_{0},f_{1},\ldots \}} {\displaystyle \{f_{0},f_{1},\ldots \}} is a sequence of orthogonal functions of nonzero _L_2-norms ‖ f n ‖ 2 = ⟨ f n , f n ⟩ = ( ∫ f n 2 d x ) 1 2 {\textstyle \left\|f_{n}\right\|_{2}={\sqrt {\langle f_{n},f_{n}\rangle }}=\left(\int f_{n}^{2}\ dx\right)^{\frac {1}{2}}} {\textstyle \left\|f_{n}\right\|_{2}={\sqrt {\langle f_{n},f_{n}\rangle }}=\left(\int f_{n}^{2}\ dx\right)^{\frac {1}{2}}}. It follows that the sequence { f n / ‖ f n ‖ 2 } {\displaystyle \left\{f_{n}/\left\|f_{n}\right\|_{2}\right\}} {\displaystyle \left\{f_{n}/\left\|f_{n}\right\|_{2}\right\}} is of functions of _L_2-norm one, forming an orthonormal sequence. To have a defined _L_2-norm, the integral must be bounded, which restricts the functions to being square-integrable.

Trigonometric functions

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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 x\in (-\pi ,\pi )} {\displaystyle x\in (-\pi ,\pi )} when m ≠ n {\displaystyle m\neq n} {\displaystyle m\neq n} and n and m are positive integers. For then

2 sin ⁡ ( m x ) sin ⁡ ( n x ) = cos ⁡ ( ( m − n ) x ) − cos ⁡ ( ( m + n ) x ) , {\displaystyle 2\sin \left(mx\right)\sin \left(nx\right)=\cos \left(\left(m-n\right)x\right)-\cos \left(\left(m+n\right)x\right),} {\displaystyle 2\sin \left(mx\right)\sin \left(nx\right)=\cos \left(\left(m-n\right)x\right)-\cos \left(\left(m+n\right)x\right),}

and the integral of the product of the two sine functions vanishes.[1] 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 \left\{1,x,x^{2},\dots \right\}} {\displaystyle \left\{1,x,x^{2},\dots \right\}} on the interval [ − 1 , 1 ] {\displaystyle [-1,1]} {\displaystyle [-1,1]} 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)} {\displaystyle w(x)} that are inserted in the bilinear form:

⟨ f , g ⟩ = ∫ w ( x ) f ( x ) g ( x ) d x . {\displaystyle \langle f,g\rangle =\int w(x)f(x)g(x)\,dx.} {\displaystyle \langle f,g\rangle =\int w(x)f(x)g(x)\,dx.}

For Laguerre polynomials on ( 0 , ∞ ) {\displaystyle (0,\infty )} {\displaystyle (0,\infty )} the weight function is w ( x ) = e − x {\displaystyle w(x)=e^{-x}} {\displaystyle w(x)=e^{-x}}.

Both physicists and probability theorists use Hermite polynomials on ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} {\displaystyle (-\infty ,\infty )}, where the weight function is w ( x ) = e − x 2 {\displaystyle w(x)=e^{-x^{2}}} {\displaystyle w(x)=e^{-x^{2}}} or w ( x ) = e − x 2 / 2 {\displaystyle w(x)=e^{-x^{2}/2}} {\displaystyle w(x)=e^{-x^{2}/2}}.

Chebyshev polynomials are defined on [ − 1 , 1 ] {\displaystyle [-1,1]} {\displaystyle [-1,1]} and use weights w ( x ) = 1 1 − x 2 {\textstyle w(x)={\frac {1}{\sqrt {1-x^{2}}}}} {\textstyle w(x)={\frac {1}{\sqrt {1-x^{2}}}}} or w ( x ) = 1 − x 2 {\textstyle w(x)={\sqrt {1-x^{2}}}} {\textstyle w(x)={\sqrt {1-x^{2}}}}.

Zernike polynomials are defined on the unit disk and have orthogonality of both radial and angular parts.

Binary-valued functions

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Walsh functions and Haar wavelets are examples of orthogonal functions with discrete ranges.

Plot of the Chebyshev rational functions of order n=0,1,2,3 and 4 between x=0.01 and 100.

Legendre and Chebyshev polynomials provide orthogonal families for the interval [−1, 1] while occasionally orthogonal families are required on [0, ∞). In this case it is convenient to apply the Cayley transform first, to bring the argument into [−1, 1]. This procedure results in families of rational orthogonal functions called Legendre rational functions and Chebyshev rational functions.

In differential equations

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Solutions of linear differential equations with boundary conditions can often be written as a weighted sum of orthogonal solution functions (a.k.a. eigenfunctions), leading to generalized Fourier series.

  1. ^ Antoni Zygmund (1935) Trigonometrical Series, page 6, Mathematical Seminar, University of Warsaw