  In this paper, we introduce the fractional Fourier series on the fractional
torus and study some basic facts of fractional Fourier series, such as
fractional convolution and fractional approximation. Meanwhile, fractional
Fourier inversion and Poisson summation formula are also given. We further
discuss the relationship between the decay of fractional Fourier coefficients
and the smoothness of a function. Using the properties of fractional Fejer
kernel, the pointwise convergence of fractional Fourier series can be
established. Finally, we present the applications of fractional Fourier series
to fractional partial differential equations with periodic boundary condition.
Moreover, we apply approximation methods on the fractional torus to recover the
non-stationary signals.
