Fourier Theory

A system is identified by its properties, and linearity, more often than not, plays a fundamental role in organizing them quantitatively. Thus, if $\psi_1$ and $\psi_2$ are two spatial amplitude profiles of a vibrating string, or two temporal histories of a simple harmonic oscillator, then so is the linear combination

$$\psi =c_1 \psi_1 +c_2 \psi_2~.$$

One says that the system obeys the linear superposition principle. Mathematically one says that the set of such $\psi$ 's forms a vector space.

By specifying functions such as $\psi(x),\psi_1(x)$ , and $\psi_2(x)$ one directs attention to the fact that (i) the system has a specific domain and that (ii) the state (or the history) of the system consists of assigning to this domain numbers that characterize the state (or the history) in numerical terms.

There are linear systems whose intrinsic properties are independent of the translations in that domain. An infinite string having constant density and tension, or a simple harmonic oscillator with its time independent spring constant and mass, or a system of differential equations with constant coefficients is a case in point. Such systems have the property that if $\psi(x)$ describes its state (or its history, in which case $x$ is the time variable), so does $\psi(x+a)$ . In other words, a linear system which is invariant under space (or time) translation has the property that $\psi(x)$ and $T_a\psi(x)\equiv \psi(x+a)$ belong to the same vector space.

This immediately raises the algebraic question: What are the eigenvalues and eigenvectors of the translation operator $T_a$ ,

$$T_a\psi(x)=\lambda_a \psi(x)~?$$

i.e. which states (or histories) have the property that

$$\psi(x+a)=\lambda_a \psi(x)~?$$

By differentiation w.r.t. $a$ one finds that the solution is

$$\psi (x)=e^{cx}, \quad$$

where $c$ is a constant. The requirement that this solution stay bounded in the whole domain $-\infty<x<\infty$ demands that the constant be purely imaginary:

$$\psi(x)=e^{ikx}~.$$

These are the tranlation eigenfunctions whose eigenvalues are $e^{ika}$ .

Fourier theory is based on introducing these functions to represent any state (or history) of the a linear translation invariant system. In brief, Fourier theory is an expression of the translation invariance of a linear sustem.

Suppose the linear system is notonly tranclation invariant, but the state $\psi$ in which it is found is periodic, i.e.

$$\psi(x+L)=\psi(x) \quad \textrm{for all }x.$$

Under such a circumstance the only translation eigenfunctions $e^{ikx}$ that can be used to describe the system are those which have the property

$$e^{ik(x+L)}=e^{ikx}\quad \textrm{for all }x.$$

This implies that

$$k=\frac{2\pi}{L}\quad n=\textrm{any integer}.$$

Thus a tranlation invariant linear system which is found in a periodic state is confined to a subspace consisting of the linear combinations of

$$e^{\frac{2\pi}{L}nx}\quad n=\textrm{any integer}.$$

A Fourier series consists of a linear combination of such eigenfunctions. The conclusion is that the theory of Fourier series is an expression of the translation invariance of a system which is in a periodic state. A ticking clock and a periodic lattice are example of such systems.