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I am reading into Gelfand Shilov spaces and in the book about distribution theory there is this space $S_{\alpha}$ defined by

$$|x^{k}\varphi^{(q)}(x)| \leqslant C_{q}A^{k}k^{k\alpha}$$

where $C_{q}$ and $A$ are constants and $k,q = 0,1,2,\dots$. Now according to the authors this definition imposes a constraint on the decrease of the fundamental functions as $|x| \rightarrow \infty$. They say that his is easily seen when dividing both sides by $|x|^{k}$ and pass to the minimum of $k$ on the right side. I don't really see that - when I do that I get

$$|\varphi^{(q)}(x)| \leqslant \frac{C_{q}A^{k}k^{k\alpha}}{|x|^{k}}$$

but what does passing to the minum mean? And how do I see the constraint?

Thanks!

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