Mathematics

I have proven before this that $f$ is integrable iff for every $\epsilon >0$ there exist step functions with $s_1\leq f\leq s_2$ such that $$\eqalign{
& \int\limits_a^b f - \int\limits_a^b {{s_1}} < \epsilon \cr
& \int\limits_a^b {{s_2}} - \int\limits_a^b f < \epsilon \cr} $$

@robjohn Now, I proved the RL lemma for step functions

So I should use that above to prove it for any integrable $f$

But I can't get to bound $$\int\limits_a^b f \left( x \right)\sin \left( {\lambda x} \right)dx$$ between the integral of step fns and $\sin \left( {\lambda x} \right)$

It is probably something really stupid, but I'm missing it.

I'd prefer you hinted me.

BenjaLim

4:02 AM

@PeterTamaroff If you know fourier series the riemann lebesgue lemma is easy

Rajesh D

4:13 AM

@PeterTamaroff First $\int\limits_a^b s_1 \left( x \right)\sin \left( {\lambda x} \right)dx$ goes to $0$ as $\lambda$ goes to infinity,late use the fact that any $f$ can be approximated with step functions, the RL follows for $f$. I had read such a proof but I am not quickly able to recollect it.

Peter Tamaroff

5:27 AM

@JasperLoy This is sad

anon

That is excellent inspiration for next April.

Peter Tamaroff

@anon =) Hey

@anon How's it going?

anon

meh

Peter Tamaroff

5:48 AM

@anon meh bad?

anon

meh is neutral or feelinglessness

Peter Tamaroff

@anon Heh well, yeah, nevermind.

@anon Did you see my RL problemup there?

anon

dunno much about that stuff

Peter Tamaroff

@anon I guess you're more into algebra, right?

anon

I never looked much at the tedious stuff in analysis

Peter Tamaroff

5:54 AM

@anon Why "tedious"?

anon

Because it involves details that bore me.

Peter Tamaroff

@anon Hehehe maybe! But one gets used to those details.

@anon What are you studing at the time?

anon

I want to get my hands dirty with some computations and look at structure constants in representation rings, and then see if I can graduate to finding ways to describe schur functors' effects. (I have found schur functosr to be a spiritual huge generalization of the newton-girard identities for symmetric polynomials via taking traces, which is what led me to them more or less.)

so IOW representation theory

Peter Tamaroff

@anon IOW?

@anon Ok, whatever those are!

@anon Is the fundamental theorem of symmetric polynomials complicated to prove?

anon

@PeterTamaroff No, it's very quick. I've paraphrased it here and here for instance.

Do you know about the NG identities at all?

Peter Tamaroff

6:04 AM

@anon Well, now I do...

@anon (I like to explain math to people that don't know about it, sometimes)

anon

The formulas in sections 2.2 and 2.4 on the NG WP page correspond to traces of the results of schur functors for the alternating and trivial representations of the symmetric groups $S_n$ respectively.

Peter Tamaroff

@anon What is a functor?

anon

@PeterTamaroff http://math.stackexchange.com/a/142103/11763

Peter Tamaroff

@anon Oooook. I'll sleep now.

@anon Bye byes.,

anon

@PeterTamaroff Roughly speaking, categories are comprised of mathematical objects and certain types of maps between them; a functor maps objects from one category into another (as well as maps, aka morphisms or arrows, from one into the other category) in such a way information like $A\xrightarrow{f} B$ entail corresponding $FA\xrightarrow{Ff}FB$.

Later.

N3buchadnezzar

6:38 AM

Hi

robjohn

9:41 AM

Why are all these people leaving?

Jayesh Badwaik

@robjohn who all? I got a -10 today from a deletion of a user.

robjohn

@JayeshBadwaik 3 have left in the last 3 days.

I lost 134 yesterday

Jayesh Badwaik

@robjohn hmm. Probably SE network can have a "Why are you leaving?" question to those who leave? This might help.

robjohn

@JayeshBadwaik For those who leave in the future. It is too late for those who've left already.

Jayesh Badwaik

@robjohn Yup. Some people can leave for no reason at all, like Jasper left Ubuntu.SE and wants his online presence clean.

robjohn

9:48 AM

@JayeshBadwaik clean? what does that mean?

Jayesh Badwaik

@robjohn They do not want to have accounts if they are not active there. This way, they don't have accounts of which they have forgotten about.

robjohn

@JayeshBadwaik That's lame. If they're not planning on going back, what does it matter if they forget about it?

I'd just make a note and file it away for later reference, if needed.

Jayesh Badwaik

@robjohn I know. But that is what people have in mind. One of my friends told me he was worried about being hacked without being known. Probably they are a little careless with their info online and hence, are afraid what info of theirs can be used if some of the accounts are hacked. I don't know, but this might be a reason. LinkedIn and Sony Hacks do not provide any reassurance.

robjohn

There is very little personal information recorded here. An email address and some IP addresses. If they want to go they can even remove the email address and their "about me" info

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$Mathematics$ Mathematics