transform, which give complete information about all the jump singularities of a function.

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May 3, 2014 at 3:50	comment	added	Rajesh D		My statement for this problem, I am trying to prove this : mathoverflow.net/q/165038/14414
May 15, 2013 at 22:26	comment	added	Rajesh D		@Igor Khavkine : I think we are no way near the solution. IMO we need a fresh outlook or idea from a totally different perspective.
Nov 12, 2012 at 14:56	comment	added	Igor Khavkine		You should concentrate on the $F_k(\xi)$ first.
Nov 12, 2012 at 13:53	comment	added	Rajesh D		we know $\hat{f}(\xi)$, but not in closed form expression though
Nov 12, 2012 at 13:48	comment	added	Rajesh D		@Igor Khvkine : I've made it clear that we assume we know $\hat(f)(\xi)$, for all $\xi \in \mathbb{R}$ except on some interval of finite length. In terms of Fourier series we assume we know all the Fourier coefficients except a finite of them. But I still do not understand how we can recover $y_i$ using the method described by you.
Nov 12, 2012 at 13:28	comment	added	Igor Khavkine		Rigor now comes down to how well you would be able to extract $F_k(\xi)$ from what you allow yourself to know about $\hat{f}(\xi)$.
Nov 12, 2012 at 13:26	comment	added	Igor Khavkine		My argument could help since it reduces your problem two the following ones: get the $F_k(\xi)$ from the $\hat{f}(\xi)$ and get solve the algebraic system $F_k(\xi) = \sum_i A_{ik} exp(i\xi y_i)$ for $A_{ik}$ and $y_i$. Both remaining problems are now algebraic equations, the first one is even linear. They are infinite dimensional, of course. However, your hypothesis that there are only finitely many discontinuities, should guarantee that there exists a finite number of $k$ and $\xi$ values to which the problem can be truncated and give the correct solution.
Nov 12, 2012 at 11:41	comment	added	Rajesh D		I mean "while I try to get a precise statement of my question it could end up with a trivial answer." Fedja has already pointed this out.
Nov 12, 2012 at 11:39	comment	added	Rajesh D		@Igor Khavkine : Thank you very much for the answer and comment. I agree that my question isn't completely well defined. But my final goal is to get there. I sincerely appreciate the answer and would like to work more on this perhaps with some help. I also understand that it may be possible that while I try to get a precise statement, the entire thing may collapse into something trivial, but thats all right as atleast I would come to a conclusion positive or negative.
Nov 12, 2012 at 11:24	comment	added	Igor Khavkine		@Rajesh, I was not trying to be rigorous in my answer. But I believe that the sketch of the argument that I gave can be turned into a rigorous one with some work, which I invite you to attempt. Of course, a rigorous theorem will depend strongly on the hypotheses you are willing to assume, and that is entirely up to you.
Nov 12, 2012 at 9:49	comment	added	Rajesh D		@Igor Khavkine : I think there is no mathematical rigour in your answer, and I'd like to know whether you agree with this.
Nov 12, 2012 at 9:47	comment	added	Rajesh D		@Igor Khavkine : We do not even know how many $y_i$ are there nor the $A_{ik}$. ALL WE KNOW is $\hat{f}(\eta)$, thats all. I am looking for a strong mathematically rigorous result in the form of a theorem with full analysis. For example if to know the jump amount, may be we should be able to create a sequence and prove it converges to the jump amount at the point of jump, not necessarily same as this but a thorough mathematical result.
Nov 11, 2012 at 17:58	history	answered	Igor Khavkine	CC BY-SA 3.0