Since the Hilbert Transform is defined as
$$
f_h(t)=P.V.\frac{1}{\pi}\int_{-\infty}^\infty\frac{f(t-x)}{x}\;\mathrm{d}x\tag{1}
$$
at any jump discontinuity, $d$, of $f$, $f_h$ blows up like $\frac{D}{\pi}\log|t-d|$, where $D$ is the size of the jump discontinuity at $d$. Furthermore, $(f^{(n)}...
ok.....i still think you may.....as it is not any expert level doubt...just that i am not able to get my mind to think in a mathematical way but still need an answer for a doubt
Well, Rajesh, we do secretly like to speak LaTeX code, don't we?
@Asaf That is an edits abuse. :) Doesn't he know that even rolling back incurs an extra edit? (Also we should warn them that $\ge 10$ edits will make the post CW.)
@Asaf Do you know why some (many?) people use \Leftrightarrow for iff, \Rightarrow for \implies and \Leftarrow for \impliedby? It always bugs me when someone doesn't use the beautiful implies sign.
TB reminds me too much of tuberculosis and T.B. would have been okay, but somehow t.b. looks nicer to my eyes. I also wanted to avoid copying J.M.'s style.
I'm gonna meet a friend for dinner soon, and the way these excursions usually end, I think I'm gonna hit the straw pretty soon afterwards. I tend to wake up with a headache usually on this particular day, so I'd prefer to know why...
@tb idioms.thefreedictionary.com/straw. I have heard it as "hit the sack" here, and I subconsciously understood what you mean. The "hit the straw" idiom is not on the list.
@JM In this case, it wasn't the Latin that was the issue. Rajesh did not understand the reference to Brutus. (In all honestly, that name sounds more like an abuse. :))
@Sri: Looks to me Mike already said what I wanted to say. If I wanted to bound the error of an integral approximated with sums, Euler-Maclaurin is always the usual candidate.
Anyway, getting back to that integral: it's already conveniently expressed as the sort of sums you'd get with the trapezoidal rule, so Euler-Maclaurin naturally presents itself (which is why one of the comments under Mike's answer mystifies me all the more).
I'm kinda bummed with this. I was going for the didactic approach, since OP had already shown that he'd tried some stuff out, and then suddenly there is a nice full answer.
Okay, it's the purpose of this site to answer questions, but I think trying to get people to think and figure it out by themselves is much more valuable than having an answer.
My take is that there should be an answer, but sometimes they appear too soon. If only we could make more of the posters post the solution, that'll be nice.
It's gonna take a whole lot of nudging from us "more experienced" users. At the very least, I've seen Arturo, Didier, and Gerry take up the campaign to nudge questioners to post the solutions they have as answers.
If we look at it from the rep viewpoint, if you post a neat solution to your own problem, an upvote will net you ten rep, as opposed to the measly two rep you get from accepting an answer.
I guess I won't fall in the "more experienced" category, but I tell new users whenever possible. But it sure is easier for me to just post that answer than to convince the OP to post the answer.
Sometimes, the OP is unsure (about the solution, whether it's ok to post it etc.) or just plain lazy. I myself overcame a significant disinclination* when I answered my own question here: math.stackexchange.com/questions/68031. (*can't think of any better word)
It wasn't the answer that was the issue, t.b. It was more a question of "should I be answering my own question?". Now I close that thought as not a real question.
Haha. I'm looking for some questions about induction on real numbers, and I ran into some question that I have answered and PEV wrote a one liner with a link to Wikipedia :-)
Thanks Asaf. With regard to the upright d, I oscillate between this side and that. It kinda looks ugly when I write a lot of these integrals strung together. But I like it when there's a lone integral somewhere. For example, I made the "d" upright in the riemann sum limit answer.
Well, I learned about typography before I learned math typography, so maybe, that's an old remnant. It cost me dearly to use \mathbf in an answer recently, but I couldn't think of a nicer way to do it.
(although I'm all for abuse of notation in such cases)
Also, after reading math books of various typographical quality, I think I've settled a bit on my "mathematical aesthetics", or whatever the term for that is...
On a different topic: sometimes I wonder who the Gravatar guys consulted for color advice. Somehow it looks like they didn't pick though their color list properly...
...such that some people end up with a puke-green or eye-melting-yellow Gravatar
(In his own words) ...an answer got accepted 31 minutes after the question was posted. This behaviour seems to be more and more common amongst certain MSE posters although it has some obvious drawbacks, to which I wish to draw attention. For example, some would-be answerers will be de facto unable to post answers until after one got accepted, if only for time zone reasons. (The present post is a good case to mention this because the accepted answer is pretty decent.)
Since the Hilbert Transform is defined as
$$
f_h(t)=P.V.\frac{1}{\pi}\int_{-\infty}^\infty\frac{f(t-x)}{x}\;\mathrm{d}x\tag{1}
$$
at any jump discontinuity, $d$, of $f$, $f_h$ blows up like $\frac{D}{\pi}\log|t-d|$, where $D$ is the size of the jump discontinuity at $d$. Furthermore, $(f^{(n)}...
