2$ by test functions and distributions

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Feb 15, 2018 at 1:22	history	edited	Michael Hardy	CC BY-SA 3.0	added 30 characters in body; edited title
May 17, 2017 at 22:02	comment	added	user21574		See my answer in mathoverflow.net/questions/150478/…
Apr 12, 2017 at 16:54	comment	added	Sayyed Hamid Banihashemi		It might also be interesting to notice that $$ lim_{ t-> \infinity } \int_0^t (sinh x/x) dx-> \infinity $$ very fast as a matter of changing from Euclidean to hyperbolic geometry.
Apr 11, 2012 at 12:29	answer	added	Bazin		timeline score: 1
May 2, 2011 at 21:25	history	edited	Alain Valette		edited tags
Jan 28, 2011 at 13:54	history	edited	Sayyed Hamid Banihashemi	CC BY-SA 2.5	added 361 characters in body; added 4 characters in body
Jan 28, 2011 at 5:48	history	edited	Sayyed Hamid Banihashemi	CC BY-SA 2.5	added 56 characters in body
Jan 27, 2011 at 22:36	comment	added	Did		@shb I could not say... You might want to provide some background.
Jan 27, 2011 at 21:49	comment	added	Sayyed Hamid Banihashemi		@Didier can't we fit the above quantum situation into a dynamical system with supersymmetry? One needs to enlarge the problem getting into QFT.
Jan 27, 2011 at 20:57	history	edited	Sayyed Hamid Banihashemi	CC BY-SA 2.5	added 553 characters in body; added 2 characters in body
Jan 27, 2011 at 20:34	comment	added	Did		@shb You mentioned supersymmetry twice (once in the post and once in a comment). I wonder what would be the analogy with the integrals in your question you have in mind.
Jan 27, 2011 at 17:55	history	edited	Ben Webster♦	CC BY-SA 2.5	added 704 characters in body
Jan 26, 2011 at 18:36	comment	added	Sayyed Hamid Banihashemi		Of course, I am using sinx/x because I am forgetful of its mysterious integral's calculus proof. There is nothing special about sinx/x except for some nice special peculiarities.
Jan 26, 2011 at 18:29	comment	added	Sayyed Hamid Banihashemi		The question is broad alright. If I knew the exact answer I would not have asked! Zen is right; you can get the answer from Mathematica too. But it seems to me that you can also do Fourier analysis on sinx/x so that the integral does not vary too much. There might be a distinction for widely varying Fourier series wiggling around sinx/x with large or unbounded integrals, and those within a given range beyond a critical limit of which the analysis breaks down. Something like supersymmetry. Now I am getting carried away. I have to think about the problem.
Jan 26, 2011 at 16:48	comment	added	Zen Harper		I don't think your actual question has much to do with your title question. Finding the (improper) integral rigorously doesn't need anything from distributions; just some elementary calculus/differential equations and simple inequalities is enough (contour integration is not needed, although it does give one nice method).
Jan 26, 2011 at 16:23	comment	added	Willie Wong		Hum, re-reading your comment and the statement of your question, I'm now not complete sure I understood you. Please give the precise meaning for the phrases "narrowing these test functions down" and "the integral falls in the interval (a,b)". I interpret the first phrase to mean "consider the subset of $C^\infty\cap L^1$" and the second to mean "so that $S_\phi \in (a,b)$". But it could also easily have meant you want to consider functions in $C^\infty_0(\pi/2 -\epsilon,\pi/2+\epsilon)$. Please clarify.
Jan 26, 2011 at 15:06	comment	added	Sayyed Hamid Banihashemi		Willie appears to be right, but it appears to me that you can do some kind of harmonic analysis on sinx/x as the range is narrowed by some epsilon of \pi/2. (I was looking at Andre's von Neumann algebras and Hilbert space comparisons)
Jan 26, 2011 at 14:12	comment	added	Willie Wong		The second question seems to be sufficiently distinct from the first one that perhaps a better thing to do is to split that off and ask it as a separate question. To better phrase your question, you should probably state what properties you want for the transformation $T$, other wise the question is very broad and open to interpretation.
Jan 26, 2011 at 14:09	comment	added	Willie Wong		For the first question, as stated, nothing much. Let $\phi \in C^\infty \cap L^1(0,\infty)$. Let $S_\phi$ be its integral against $\sin x / x$, which we assume to be non-zero. Consider the test function $\psi = \frac{\pi}{2 S_\phi} \phi$, then $S_\psi = \pi/2$ exactly. So you only rule out test functions which are "$L^2$ orthogonal" to $\sin x / x$.
Jan 26, 2011 at 14:05	comment	added	André Henriques		+1 for spending the time and effort to reformulate your question nicely.
Jan 26, 2011 at 14:01	history	edited	Sayyed Hamid Banihashemi	CC BY-SA 2.5	added 2 characters in body; edited title; deleted 4 characters in body; added 2 characters in body
Jan 26, 2011 at 13:56	history	edited	Sayyed Hamid Banihashemi	CC BY-SA 2.5	edited body; added 10 characters in body; edited title
Jan 26, 2011 at 13:35	history	asked	Sayyed Hamid Banihashemi	CC BY-SA 2.5

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