  It is a classical theorem that if a function is integrable along the boundary
of the unit circle, then the function is the nontangential limit of a
holomorphic function on the open disc if and only if its Fourier coefficients
for nonnegative integers are zero. In this article we generalize this result to
higher complex dimensions by proving that for an integrable function on the
unit sphere, it is ``boundary trace'' of a holomorphic function on the open
unit ball if and only if two particular families of integral equations are
satisfied. To do this, we use the theory of Hardy spaces as well as the
invariant Poisson and Cauchy integrals. This article is written in a style that
is meant to be welcoming to those who have taken a course in complex analysis
but who are not necessarily experts in the field.
