  We prove that if a multiple trigonometric series is spherically Abel summable
everywhere to an everywhere finite function $f(x)$ which is bounded below by an
integrable function, then the series is the Fourier series of $f(x)$ if the
coefficients of the multiple trigonometric series satisfy a mild growth
condition. As a consequence, we show that if a multiple trigonometric series is
spherically convergent everywhere to an everywhere finite integrable function
$f(x)$, then the series is the Fourier series of $f(x)$. We also show that a
singleton is a set of uniqueness. These results are generalizations of a recent
theorem of J. Bourgain and some results of V. Shapiro.

Classical Analysis and ODEs