Suppose we have a ball $B \subset \mathbb R^n$ and $\alpha >0$, and $\{ u_j\}_{j\geqslant-1} $ is a sequence of smooth functions such that the Fourier transforms $\mathcal{F}(u_j) $ are supported in $2^jB$, and $\lVert u_j\rVert_{L^\infty}\leqslant C2^{-j\alpha}$ for all $j$, where $C$ is a constant independent of $j$. How to show the series $$\sum_{j\geqslant-1}u_j$$ converges in the space of tempered distributions?
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$\begingroup$ Hans Triebel studied problems of this kind some time ago: I am not sure if this could be of some help to you, but perhaps a look at his monograph The Structure of Functions is worth doing, at least for the first chapter. $\endgroup$– Daniele TampieriDec 12, 2021 at 6:57
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The following arguments maybe a solution. For any Schwartz function $f\in \mathcal{S}$, we can estimate $$|<u_j,f> |\leqslant \|u_j\|_{L^\infty}\|f\|_{L^1}\leqslant C 2^{-j\alpha}\|f\|_{L^1}$$which concludes that $$<\sum_{j=-1}^{\infty} u_j,f>$$converges.