Spectral Resolution of a Function

(or spectral representation of a function), an expansion of a function in a series of eigenfunctions of some linear operator (for example, a finite-difference, differential, or integral operator) acting in a function space or a generalization of such an expansion. Special cases of a spectral resolution are the expansion in a Fourier series of a function defined on a closed interval—that is, the harmonic analysis of oscillations—and expansions in terms of other familiar complete systems of functions.

In the case of a linear operator A with a continuous spectrum, eigenfuiictions, in the ordinary sense, do not exist. Nevertheless, it is often possible to determine the eigenfunctions, provided that we do not require them to be elements of the function space in which A acts. The spectral resolution of a broad class of functions can then be defined as a representation in terms of an integral over a system of functions of a continuously varying independent variable. Fourier integral representation is an example of this type of spectral resolution. When the operator A is not self-adjoint, it is necessary to consider, besides eigenfunctions, chains of functions associated with the eigenfunctions. In many cases, a completeness theorem for the system of all eigenfunctions and associated functions can be proved for such operators in function spaces. On this basis, spectral resolutions of a broad class of functions with respect to all possible eigenfunctions and associated functions of A can be obtained.

Spectral resolutions of functions are widely used for the solution of a variety of finite-difference, differential, and integral equations. Such resolutions have many applications in classical mechanics (particularly vibration theory), electrodynamics, quantum mechanics, communication theory, automatic control theory, and other branches of mathematical physics and applied mathematics.