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?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.