#
Fourier Theory on the Complex Plane IV

Representability of Real Functions

by their Fourier Coefficients

**Jorge L. deLyra**

* Department of Mathematical Physics,
Physics Institute,
University of São Paulo *

**May 4, 2015**

### Abstract:

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.