Orthogonal (series)

In mathematics, particularly in functional analysis and Fourier analysis, the term "orthogonal series" refers to an infinite series of the form:

∑ c_n φ_n(x)

where:

• c_n are constants (often real or complex numbers), called the coefficients of the series.
• φ_n(x) are functions of a variable x, which are members of an orthogonal set of functions. This means that the integral of the product of any two distinct functions in the set over a specified interval is zero. Formally, if [a, b] is the interval under consideration, then:

∫_a^b φ_m(x) φ_n(x) w(x) dx = 0 for m ≠ n

where w(x) is a non-negative weight function. If w(x) = 1, the functions are simply called orthogonal. If the integral equals 1 when m = n, the functions are called orthonormal.

The concept of orthogonality is a generalization of the geometric notion of perpendicularity to functions. Orthogonal series provide a way to represent functions as a linear combination of simpler, orthogonal basis functions.

The convergence properties of an orthogonal series, and the properties of the function to which it converges (if it converges), depend on the specific orthogonal set of functions used, the coefficients c_n, and the weight function w(x). Commonly encountered orthogonal series include Fourier series (using trigonometric functions) and series of orthogonal polynomials (e.g., Legendre polynomials, Chebyshev polynomials).

The coefficients c_n are often determined using the orthogonality property of the functions φ_n(x). Given a function f(x) that we wish to represent as an orthogonal series, the coefficients can be found by projecting f(x) onto each of the basis functions. This is achieved by multiplying f(x) by φ_n(x)w(x) and integrating over the interval [a, b]:

c_n = (∫_a^b f(x) φ_n(x) w(x) dx) / (∫_a^b φ_n(x)² w(x) dx)

This formula relies on the orthogonality property and allows us to decompose a function into its components with respect to the orthogonal basis. The denominator normalizes the basis functions. If the basis functions are orthonormal, the denominator equals 1.

The study of orthogonal series is fundamental in many areas of mathematics, physics, and engineering, including signal processing, image analysis, and the solution of differential equations.