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Hi Rajesh,
The wording of your question is quite convoluted. Probably this is because you are an expert in a language which is not mathematics (I see that you are a PhD student in Signal Processing), and you are having to produce a translation on the fly. Let me try to rewrite your question, to confirm that I understand it.
You consider the class of real-valued functions $f$ on the circle $S^1 = \mathbb R / \mathbb Z$ for which: (1) the set of points at which $f$ is not infinitely differentiable is countable and dense; (2) let $x$ be a point at which $f$ is $k$-times differentiable but not $(k+1)$-times, then both the left and right limits of the $(k+1)$th derivative of $f$ exist at $x$, and fail to agree. As far as I can tell, your notation is superfluous, and you never use the function you call $ck$. Oh, and you also request that $f$ be right-continuous at $0$.
Finally, you claim that this class of functions is a vector space, and you ask whether it is closed under convolution in $S^1$. Note that as written it definitely is not a vector space: the constant function $0$ fails condition $1$. But presumably you mean to weaken this?
It is not automatically clear to me that the convolution is even defined, but this is probably covered by your somewhat strict regularity requirements. Almost certainly the argument for closure under convolution is going to go something like this: by condition (2), the set of points at which any particular function fails to be $k$-times differentiable is discrete and hence finite; then the set of points at which the convolution of $f$ and $g$ fails to be regular is made out of those points at which $f$ and $g$ are each unregular, and you can do a direct analysis. This will be easier with the correct weakening in the preceding paragraph --- then almost certainly your class will be naturally filtered by easy conditions.
In order for this question to be suitable at MO, maybe most important is to improve the statement of the question. (Don't forget to look over http://mathoverflow.net/howtoask .) Currently you take three pages to say what could fit in less than one. I would also be interested to hear why you are interested in this question. But do give math.stackexchange a bit longer to provide an answer.
I worry that even with improvements in the statement, the question still will not be MO-appropriate. If my understanding of the question is correct, it is at an undergraduate-homework level. I tend to think graduate-level-in-math homework can be appropriate at MO, although there is disagreement as to this. And I understand that this question is not, in fact, undergraduate homework. But it's at that mathematical level.
I hope this helps,
Theo
Hi Rajesh,
(0. There are a couple "Theo"s on MO, and I'm not on M.SE, out of laziness. So probably you know a different "Theo"?)
I agree that you mention the function "ck" in the definition. My point was that you don't need to. As far as I can tell, your definition is: given f, define Cf as the counting the number of times f is differentiable at a given point. Now set ck = Cf. Now consider those functions for which ck = Cf has some property.
Probably I was skimming too fast. Your set S_pc is the union of your set S_p with the constant functions, and this certainly includes 0. It would then be better to never mention the set S_p, and define the set of functions that you want as being of some shape, or being constant.
Let me mention here that it's worth practicing and editing mathematical writing, because shorter writing is easier to skim.
I mean only that convolution involves integration, and not all functions are integrable. I'm not a real analyst, and so would like you to include one sentence for why your functions are integrable.
By "weakening" I meant that I thought the proceeding paragraph was incorrect, because the claim made was too strong.
Hope that helps, Theo
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