  In this paper, a Fourier series in fractional dimensional space is introduced
for an arbitrarily periodic function $f(t;\alpha)$. We call it fractional
Fourier series of the order $\alpha$. Extending the basis functions of the
linear space into fractional one, by rotation transformation, we define a real
and complex Fourier series and obtain their coefficients. It is also shown that
the fractional derivative of a periodic function can be realized through
(fractional) Fourier series with modified coefficients.
