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I had asked two specific questions on "mse" but haven't got any answer for my questions. The first question is this for which I even offered a bounty of $50$. The bounty period expired but no answers came. The second question is here. What do I do to get an answer for this question. Does no answer mean that nobody knows about this specific area or something else ? Thanks..

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Okay, I've looked a bit at your questions, and here are my thoughts.

  • How to construct examples of functions in the Spaces of type $\mathcal{S}$

    I guess it's clear that Gelfand-Shilov spaces aren't exactly a commonly known concept. Because of this it is incumbent on you to make sure that people without specific knowledge of these spaces (but sufficiently well-versed in related matters) can fumble their way to the answer without too much reference checking. This is well out of my area of expertise, but in doing some searching I came across the following article

    • S.J.L. van Eijndhoven, Functional analytic characterizations of the Gelfand-Shilov spaces $S_α^β$, Indagationes Mathematicae (Proceedings) vol.90, no.2, pp.133-144, doi

    where the following definition of $S_\alpha$ ($\alpha \geq 0$) is given

    The space $S_\alpha$ consists of all functins $\phi \in S$ which satisfy $$\exists_{A > 0} \forall_{l \in \mathbb{N}_0} \exists_{B_l > 0} \forall_{k \in \mathbb{N}_0} : \sup_{x \in \mathbb{R}} | x^k \phi^{(l)}(x) | \leq B_l A^k ( k!)^\alpha.$$

    Further, the article mentions

    The Schwartz' space $S$ consists of all $C^\infty$-functions $\phi$ on $\mathbb{R}$ with the property that $\sup_{x \in \mathbb{R}} | x^k \phi^{(l)} (x) | < \infty$ for all $k,l \in \mathbb{N}_0 := \mathbb{N} \cup \{ 0 \}$.

    I can only assume that this is close to what you are talking about when you define

    • $\mathcal{S}_\alpha: |x^k\varphi^{(q)}(x)|\le C_qA^kk^{k\alpha}\qquad (k,q=0,1,2,...)$ ... where the constants $A,C_q$ depend on $\varphi$.

    But even then the exact quantification is left up in the air.

    So as a first step, include full and clear definitions of the uncommon concepts in your question. If you came across these spaces studying a specific text, you should mention what text that is.

    As you have formatted the question it is not easy to determine where exactly you ask a question. Also, as phrased your question

    In short I want to know for which values of $\alpha\:,\:\beta$, functions (say $e^{-|x|^2}$ ) belong to these spaces.

    may be considered too broad if interpreted in one way (that $e^{-|x|^2}$ is only one function that you might be interested in, but would welcome answers for any $S$-function).

  • Laplace transform of functions related to type $\mathcal{S}$, and the relation to entire functions

    First of all, this question is actually two largely unrelated questions which should be asked separately.

    The title of your question is very nondescriptive and wouldn't cause me, for one, to look at it even if tagged with any of my favourite tags.

    Again we have the problem of undefined notions which aren't very common.

    You state an assertion, but you should probably source that assertion: where did you read it? (Title, author, etc.)

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  • $\begingroup$ Thank You very much for your reply. I have edited the first question and tried to make it more precise based on your guidelines. I will soon edit the second question as well. $\endgroup$ – Hirak Aug 26 '15 at 6:27
  • $\begingroup$ I have edited the second question as well !! Does this make things more coherent and clear ? $\endgroup$ – Hirak Aug 27 '15 at 7:35
  • $\begingroup$ @Arthur I have a same problem. There is a question which is unsolved. what should I do ? Thanks $\endgroup$ – Cardinal Aug 27 '15 at 10:13
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    $\begingroup$ @ArthurFischer Why don't post the math part of this answer on M.SE where it belongs? $\endgroup$ – user26857 Aug 28 '15 at 22:34
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"Does no answer mean that nobody knows about this specific area or something else?"

Quite possibly, You had to create a tag for your first post. This signals that the subject you are interested in doesn't have a large following here. For instance, fractal analysis is a weak tag. You could also take a look at the new tags page to get a feel for the various subjects on MSE that are unpopular.

In addition, you should look closely at the popular tags on MSE. PDEs are on the bottom of the second page. By the Pareto Principle, the majority of the attention to answering a topics questions will be concentrated in the minority of topics.

"What do I do to get an answer for this question"

You can always migrate your question to Math Overflow if you feel that the question is sufficiently difficult to warrant a professional comb-over.

In addition, I'd take advantage of the Pareto Principle. If you can phrase your question so that it relates in a tangible way with other, more popular, subjects, you can add tags that are more likely to be answered.

However, I'd advise you don't add more than one bounty, based another question I looked into here.

Instead, I'd suggest you try breaking up your question into smaller chunks. If there are 5 manageable questions rather than 1 challenging question, you'll have better luck having at least having part of your problem solved.

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    $\begingroup$ +1 for that last paragraph. In my short experience, that is spot on for mathematical research in general. Don't prove a single hard theorem, break it into five consecutive lemmas, each of which has a two liner proof. $\endgroup$ – Asaf Karagila Aug 19 '15 at 15:55
  • $\begingroup$ @Zach466920, I like the approach of your answering this question. I will see what I can do with the questions. $\endgroup$ – Hirak Aug 20 '15 at 4:45
  • $\begingroup$ I think the questions asked are straight forward to understand. I went back and checked. Is there specific thing which I can add further to make it more tangible ?? $\endgroup$ – Hirak Aug 20 '15 at 7:44
  • $\begingroup$ @loophole relate it to other subjects. If you have subject specific definitions share those definitions in terms other people can understand. There sadly isn't a "quick" fix to the issue. $\endgroup$ – Zach466920 Aug 20 '15 at 13:25

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