Convergence of the fourier series for differentiable functions of a multidimensional $p$-adic argument

This article discusses the convergence of the Fourier series for functions of the multidimensional $p$-adic argument. For this purpose we define the multidimensional Mahler function and partial sums of Fourier series for the functions of multidimensional $p$-adic argument. We calculate the norm of the $m$-th derivatives of multidimensional Mahler functions and prove the criterion of $m$ times continuously differentiability in terms of Mahler coefficients. We represent coefficients and partial sums of multidimensional Fourier series in terms of coefficients and partial sums of one-dimensional Fourier series. The main result states that for positive integers $m \ge n$ the Fourier series for function $C^m(\mathbb{Z}_p^n)$ converges uniformly. An example of $f \in C^{n-1}(\mathbb{Z}_p^n)$ with divergent Fourier series is given.