(Definition)
Let $m, n$ be positive integers with $m \le n.$
Let $f:\Bbb R^n \rightarrow \Bbb C$ be a fucntion in $n$ variables $x_1, x_2, \ldots , x_n$.
Suppose that $f$ is $1$-periodic in $x_1, \ldots, x_m$, i.e., $$f(x_1 + c_1,\ldots,x_m + c_m, x_{m+1},\ldots, x_n) = f(x_1,\ldots,x_n) $$ for $(c_1,\ldots,c_m) \in \{0,1\}^m.$
Let $$a(k_1,\ldots,k_m;x_{m+1},\ldots,x_n)=\int_{0}^1 \cdots \int_{0}^1 f(x_1,\ldots,x_n) \mathrm{exp}(-2 \pi i (k_1 x_1 + \cdots + k_m x_m)) dx_1 \cdots dx_m $$ be the Fourier coefficients of $f$.
(Problem)
I would like to know when the Fourier series $$F(f) = \sum_{k_1,\ldots,k_m \in \Bbb Z} a(k_1,\ldots,k_m;x_{m+1},\ldots,x_n) \mathrm{ exp }(2\pi i(k_1 x_1 + \cdots + k_m x_m)) $$ of $f$ converges absolutely to $f$ (because the summation may be multi-dimensional, I impose absolute convergence). To be concrete, I have two problems: (1) When does $F(f)$ converge absolutely to $f$? (2) When does $F(f)$ converge absolutely to $f$ and $$ \sum_{k_1,\ldots,k_m \in \Bbb Z} \lvert a(k_1,\ldots,k_m;x_{m+1},\ldots,x_n) \mathrm{ exp }(2\pi i(k_1 x_1 + \cdots + k_m x_m)) \rvert = \sum_{k_1,\ldots,k_m \in \Bbb Z} \lvert a(k_1,\ldots,k_m;x_{m+1},\ldots,x_n) \rvert $$ converge uniformly on any compact subsets in $\Bbb R^{n-m}$?
If $m=n=1$, the condition for convergence is well know. But I cannot find any textbook dealing with the general case. I guess that (1) and (2) hold if $f$ is smooth ($C^{\infty}$-function), but I cannot prove it. Please tell me any sufficent condition for the problem (1) or (2). If you know a good reference about this problem, please tell me as well.