  A correspondence between arbitrary Fourier series and certain analytic
functions on the unit disk of the complex plane is established. The expression
of the Fourier coefficients is derived from the structure of complex analysis.
The orthogonality and completeness relations of the Fourier basis are derived
in the same way. It is shown that the limiting function of any Fourier series
is also the limit to the unit circle of an analytic function in the open unit
disk. An alternative way to recover the original real functions from the
Fourier coefficients, which works even when the Fourier series are divergent,
is thus presented. The convergence issues are discussed up to a certain point.
Other possible uses of the correspondence established are pointed out.
