  The Fourier-based analysis customarily employed to analyze the dynamics of a
simple pendulum is here revisited to propose an elementary iterative scheme
aimed at generating a sequence of analytical approximants of the exact law of
motion. Each approximant is expressed by a Fourier sum whose coefficients are
given by suitable linear combinations of Bessel functions, which are expected
to be more accessible, especially at an undergraduate level, with respect to
Jacobian elliptic functions. The first three approximants are explicitely
obtained and compared with the exact solution for typical initial angular
positions of the pendulum. In particular, it is shown that, at the lowest
approximation level, the law of motion of the pendulum turns out to be
adequately described, up to oscillation amplitudes of $\pi/2$, by a sinusoidal
temporal behaviour with a frequency proportional to the square root of the
so-called "besinc" function, well known in physical optics.
