Orthogonal System of Functions

a system of functions {ϕ_n(x)}, n = 1, 2, …, that are orthogonal with respect to some weight function ρ(x) on the interval [a, b]; that is.

For example, the trigonometric system 1, cos nx, sin nx, for n = 1, 2, …, is an orthogonal system with weight 1 on the interval [–π, π]. The Bessel functions Jv(μ_n^vxl), where n = 1, 2,…, and the μ_n^v are the positive zeros of J_v (x), form for each v > –½ an orthogonal system with weight x on the interval [0, l].

If every function ϕ (x) of an orthogonal system of functions satisfies the condition

then the system of functions is said to be normalized. Any orthogonal system of functions can be normalized by multiplying ϕ_n(x) by the normalizing factor .

The systematic study of orthogonal systems of functions began with Fourier’s method of solving boundary value problems in mathematical physics. This method yields, for example, solutions of the Sturm-Liouville problem for the equation [ρ(x)y′]′ + q(x)y = λy that satisfy the boundary conditions y(a) + hy′(a) = 0 and ẏ (b) + Hy′(b) = 0, where h and H are constants. These solutions, the eigenfunctions of the problem, form an orthogonal system of functions with weight ρ(x) on the interval [a, b].

In his research on interpolation by the method of least squares and on the problem of moments P. L. Chebyshev discovered an extremely important class of systems of orthogonal functions—the class of systems of orthogonal polynomials. In the 20th century, research on orthogonal systems of functions has been based chiefly on the theory of the Lebesgue integral and Lebesgue measure and has become an independent branch of mathematics. A fundamental problem of the theory of systems of orthogonal functions is the expansion of a function f(x) in a series of the form ∑C_nϕ_n(x), where {ϕ_n(x)} is an orthogonal system of functions. If we put formally f(x) = ∑C_nϕ_n (x), where {ϕ_n(x)} is a normalized system of orthogonal functions, and allow term-by-term integration, then by multiplying this series by ϕ_n (x)ρ(x) and integrating from a to b, we obtain

The coefficients C_n, which are called the Fourier coefficients of the function with respect to the system {ϕ_n(x)}, have the property that the linear combination

best approximates this function in the mean. In other words, the mean square error

has the smallest value compared with the value associated with any other linear combinatior

for the same n. This fact implies, in particular, the Bessel inequality

The series

where the coefficients C_n are computed from formula (*), is called the Fourier series of f(x) with respect to the orthonormal system {ϕ_n(x)}. Of prime importance for applications is the question of whether the function f(x) is uniquely determined by its Fourier coefficients. Orthogonal systems of functions for which this is the case are called complete. The conditions for the completeness of an orthogonal system of functions can be given in several equivalent forms. (1) Any continuous function f(x) can be approximated in the mean to any desired degree of accuracy by linear combinations of the functions ϕ_k (x); that is,

The series

is then said to converge in the mean to the function f(x). (2) For any function f(x) that is square integrable with respect to the weight p(x), the Liapunov-Steklov (Parseval) completeness condition

is satisfied. (3) There does not exist a nonzero function that is square integrable on the interval [a, b] and is orthogonal to all the functions ϕ_n(x), n = 1, 2, ….

If square integrable functions are viewed as elements of a Hilbert space, then orthonormal systems of functions are systems of basis vectors for the space, and expansion in a series of orthonormal functions is the expansion of a vector in terms of the basis vectors. With this approach, many concepts of the theory of orthonormal systems of functions acquire an obvious geometric meaning. For example, formula (*) says that the projection of a vector on a basis vector is equal to the scalar product of the vector and that basis vector. Moreover, the Liapunov-Steklov (Parseval) equality can be interpreted as the Pythagoras theorem for an infinite-dimensional space—that is, the square of the length of a vector is equal to the sum of the squares of its projections on the coordinate axes. Completeness of an orthogonal system of functions means that the smallest complete sub-space containing all the vectors of the system coincides with the entire space. Other examples could easily be given.