§2 Topology and harmonic analysis.

§ 2.2. Functional analysis and topology.

Physics describes objects by functions on appropriate domains. Functions on a manifold {\cal M} must respect its topology.

Example 1c: Functional analysis on the circle. We model the circle by the interval J=\{-1/2,1/2\} on the real line. Consider on R^{1} all exponentiaL functions

f_{k}(x)=\exp(ikx),\:-\infty<k<\infty. (9)

Their values will be unique on the domain J provided they are periodic, that is

f_{k}(x+1)=\exp(ik(x+1)):=f_{k}(x)\exp(ik)=f_{k}(x). (10)

Periodicity requires k=k_{n}=n\: 2\pi,n=0,\pm 1,\pm 2,..., and so restricts the values of k to the discrete subset k_{n}. The discrete subset f_{{k_{n}}}(x) of functions on the interval J=\left[-\frac{1}{2},\frac{1}{2}\right] fulfills the homotopic boundary condition f_{{k_{n}}}(1/2)=f_{{k_{n}}}(-1/2).

Repetition of the discrete translation step eq.10 generates the discrete translation or infinite cyclic group C_{{\infty}}. The periodic set \exp(ik_{n}x) is the basis of complex Fourier series analysis of a periodic function. It represents any periodic function as an infinite series with complex coefficients a_{n}. These can be found by integration over the interval J:

\displaystyle f(x):f(x+1)=f(x), (11)
\displaystyle f(x)=\sum _{{n=0,\pm 1,\pm 2,...}}a_{n}\exp(ik_{n}x),
\displaystyle a_{n}=\int _{{-1/2}}^{{1/2}}dx\: f(x)\exp(-ik_{n}x)

We conclude that the particular functions \exp(ik_{n}x),\: n=0,\pm 1,\pm 2,... form a basis of all functions on the circle S^{1} and, when extended from the interval J to the real line, of all periodic functions on the line.

A useful and constructive characterization of the basis of Fourier analysis arises in terms of the deck transformation group: Any basis function is invariant under the action of the discrete translation group H={\rm deck}(S^{1}), which in turn is isomorphic to the fundamental group C_{{\infty}} of the circle. In case of the circle it suffices already to require the invariance under the generator of the translation group, that is to require periodicity eq. 11.