On "uniqueness" of Fourier expansion It is well known that Taylor series is unique, i.e., if
$$
\sum_{n_1, \ldots, n_d \ge 0} a(n_1, \ldots, n_d)x_1^{n_1} \cdots x_d^{n_d} = 0 
$$
holds around $(x_1, \cdots, x_d)=0$ and the left hand side converges absolutely, then
$ a(n_1, \ldots, n_d) = 0 $
for all $ n_1, \ldots, n_d \ge 0$. 
I am interested in the case of Fourier series.
$\mathbf {Definition}$
Let $d \in \mathbb Z_{\gt 0}$, $n = (n_1, \ldots, n_d)$ be a variable in $\mathbb Z^d$ and $a(n) \in \mathbb C$.
Let $X$ be an order of summation of $\sum_{n \in \mathbb Z}$.
Let 
$$
\mathscr F _X (a, x) = \sum_{n \in \mathbb Z^d}  a(n) \mathrm {exp} (2 \pi i(n \cdot x))
$$
be a convergent Fourier series summed up in the order $X$, where $x \in \mathbb R^d$ and $n \cdot x$ is the standard inner product.
$\mathbf {Question}$
Suppose that $\mathscr F_X(a,x) = 0$. Then, can we conclude that $a(n) = 0$ for all $n \in \mathbb Z^d$ ?
Of course, if X is the standard (cubic) summation (i.e., $\lim_{N \to \infty} \sum_{-N \le n_1, \ldots n_d \le +N}$) and $\mathscr F_X(a, x) $ converges uniformly in $x$ on compact sets, then $a(n) = 0$ because 
$$a(n) = \int _{x \in {[0,1]}^d} \mathscr F_X(a,x) \mathrm{exp}(-2 \pi i n \cdot x) dx$$
are the Fourier coefficients of $\mathscr F_X(a,x) = 0$. On the other hand, if $\mathscr F_X(a, x)$ does not converge uniformly, we may have a chance that $a(n) \ne 0$ for some $n \in \mathbb Z^d$. But I do not have any example.
If anyone knows a hint, an example or a good reference, please tell me.
 A: Interesting you should raise this question; trying to give a definitive answer to this question is what led Georg Cantor to develop set theory (this is laid out in a good biography of Cantor by J. W. Dauben.
Cantor's final result:Let the zero function be represented by a Fourier series within a given interval.  Let there be a set of points in the interval (possibly empty) called the exception set; we assume the series converges in the interval except on this set, where we make no assumption (it may converge, it may not).  The exception set may have accumulation points; call the set of accumulation points the first derived set.  This set may itself have accumulation points; call them the second derived set.  And third, fourth, and so on.
THEOREM: Suppose the N'th derived set, for some finite N, is empty.  Then the representation of the zero function on that interval is unique; i.e, all Fourier coefficients must be zero.
THEOREM: Suppose the N'th derived set is never empty for any finite N.  Then the representation of the zero function on that interval need not be unique.
I don't know exactly how this extends to multi-dimensional space, but I suspect the relevant theorems are essentially the same.
