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$\mathbb K$ is a $\textit{local field}$ if it is any totally disconnected, locally compact, non-discrete, complete field.For examples: $\mathbb Q_p$, any finite extension of $\mathbb Q_p$, field of Laurent series on finite field. For each $\gamma\in\mathbb Z$, we denote $B_{\gamma}=\{x\in\mathbb K: \; |x|\leq q^\gamma\}$, the usual balls.

Let $f$ be any function in $L^2(\mathbb K)$ and $\chi_\gamma$ is the chareteristic function of $B_\gamma$. It is obvious that $f_\gamma=f\cdot \chi_\gamma$ converges pointwise to $f$ and $f_\gamma\to f$ in $L^2$ when $\gamma\to-\infty$. Is it true that their Fourier transform $Ff_n$ also converges pointwise almost everywhere to $Ff$

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