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I am asking about the same question as this post: Translation of a Schwartz function is a Schwartz function?

Namely: If $f\in S(\mathbf{R}^n)$ and for all $y\in\mathbf{R}^n$ then $\tau_{y}f\in S(\mathbf{R^n})$? Equivalently, is it true that we have $\sup_{x \in \mathbf{R}^N} |x^{\alpha} \partial_{x}^{\beta} \tau_{y}f(x)|<\infty$?

I am not very familiar with the multi index Binomial Theorem and it seems like the result follows from there. Can someone elaborate further why exactly this is true?

Mathematics_Beginner
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    Can you prove this in the very simple one-dimensional case? The general case is just excess notation. – peek-a-boo Apr 22 '22 at 02:15
  • @peek-a-boo I can see why in the one dimensional case is true. I think I am having trouble thinking about the multi index case because we are not sure if the vector of the shift has all positive terms. So when taking the multi-index power, we will potentially get an oscillating series, wouldn't we? – Mathematics_Beginner Apr 22 '22 at 02:18
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    $\alpha, \beta$ are multindices of non-negative integers, so I'm not sure what you mean. Again, if this calculation is not clear, you should try it first with a concrete choice of multindices, maybe in dimension 2 say. The general case really is just a notational change. – peek-a-boo Apr 22 '22 at 02:24

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\begin{align}\sup_x |x^\alpha D^\beta f(x-y)| &= \sup_x |(x+y)^\alpha D^\beta f(x)| \\ &\le \sup_x \sum_{\gamma\le \alpha} C_{\gamma,\alpha}|x|^{|\gamma|} |y|^{|\alpha-\gamma|} |D^\beta f(x)| \\ &\lesssim_{\alpha,y} \sum_{\gamma\le \alpha} [f]_{\gamma,\beta} < \infty \end{align} where $[f]_{\alpha,\beta}$ is the seminorm $[f]_{\alpha,\beta} = \sup_{x} |x^{\alpha}D^\beta f(x)|$.

Calvin Khor
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  • So is $\gamma$ also a multi-index? I am not sure what it means here when you wrote $\gamma \leq \alpha$ since both of them are in $\mathbf{N}_0^N$? – Mathematics_Beginner Apr 22 '22 at 03:02
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    @Mathematics_Beginner yes, it means that $\gamma_i \le \alpha_i$ for all $i=1,\dots,N$. Same notation as https://en.wikipedia.org/wiki/Binomial_theorem#Multi-binomial_theorem, and you can find the rest of multi-indices notation here – Calvin Khor Apr 22 '22 at 03:14
  • Thank you so much for the clarification! What does it mean when you write $|x|^\gamma$ as in this case we would have $|x| \in \mathbf{R}$ and not $\mathbf{R}^N$ anymore, right? I haven't seen multi index applied onto real numbers before. Moreover, is the semi-norm you wrote down here equivalent to the semi-norm $\sup_x |x^\alpha D^\beta f(x)|$? – Mathematics_Beginner Apr 22 '22 at 03:27
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    Right, that was me being sloppy. It should be $|x|^{|\gamma|}$, sorry. So yes, to the second question – Calvin Khor Apr 22 '22 at 03:28