  The results presented in this paper are refinements of some results presented
in a previous paper. Three such refined results are presented. The first one
relaxes one of the basic hypotheses assumed in the previous paper, and thus
extends the results obtained there to a wider class of real functions. The
other two relate to a closer examination of the issue of the representability
of real functions by their Fourier coefficients. As was shown in the previous
paper, in many cases one can recover the real function from its Fourier
coefficients even if the corresponding Fourier series diverges almost
everywhere. In such cases we say that the real function is still representable
by its Fourier coefficients. Here we establish a very weak condition on the
Fourier coefficients that ensures the representability of the function by those
coefficients. In addition to this, we show that any real function that is
absolutely integrable can be recovered almost everywhere from, and hence is
representable by, its Fourier coefficients, regardless of whether or not its
Fourier series converges. Interestingly, this also provides proof for a
conjecture proposed in the previous paper.
