Discussion between infinity and JacobsonRadical

Feeds

07:24

A: How to show $\|e^{ikx}-e^{ix}\|_{L^{1}}$ does not converge to $0$ when $k\rightarrow\infty$?

Continuoing your last equation: $\int_{-\pi}^{\pi} |e^{iy}-1| dy = \int_{-\pi}^{\pi} |cos(y) +isin(y) -1|dy =\int_{-\pi}^{\pi}(cos(y)-1)^2 +sin(y)^2)dy =\int_{-\pi}^{\pi} 2cos(y)+1 dy = 2\pi$

JacobsonRadical

Thank you so much! Perfect answer.

infinity

@Mindlack oops. you are right.. i will try to fix it.

JacobsonRadical

oh I also found that the last equality you should have $\int_{-\pi}^{\pi}2\cos(y)$, instead of $+1$.

infinity

@JacobsonRadical also right.. sorry.

JacobsonRadical

@infinity nonon, its fine. let's work it out.

infinity

07:24

ok so we have $\sqrt(2) \int_{-\pi}^{\pi}\sqrt{cos(y)}dy = 2\sqrt{2}\int_0^{\pi}\sqrt{cos(y)} dy$

JacobsonRadical

@infinity then wolfram alpha said the integral is not zero, but how could we prove it?

here I come

infinity

Hi. do you use render mathjax ? so we can write here equation in the '$'

JacobsonRadical

no..

let me check what it is...

it seems like this one is a coding source..

can I directly apply it to the web?

infinity

math.ucla.edu/~robjohn/math/mathjax.html

add it to your bookmarks

and click on it when i write $\int $

(add the render mathjax)

JacobsonRadical

then let me go to Google Chorme

one minite

Safari doesn't work

infinity

07:29

oh ok.

JacobsonRadical

..well okay.... it seems like I forgot my password or something

let us begin chatting

I can understand the '$'

infinity

so the integral is $\int \sqrt{2-2cos(y) }dy$

JacobsonRadical

(okay, I make it work on safari)

yes. the integral is this

infinity

so you can see math equations now here?

and this can be $2\sqrt{2}\int_0^{\pi} \sqrt{1-cos(y)}dy$

JacobsonRadical

yes

no problem

infinity

07:33

because the function in the integral is symmetric(f(-y)=f(y) )

JacobsonRadical

yes

oh okay

$\cos(y)\leq -1$

so $1-\cos y\geq 2$

infinity

ok and the last integral we can calculate directly

JacobsonRadical

and thus our equation $\geq 2\sqrt{2}\times\sqrt{2}\times 2\pi$

what am I saying?!?!?!?!?

never mind. this is midnight, I am crazy

infinity

yeah it happens to everyone :P

JacobsonRadical

why could we compute it?

infinity

07:35

but everything is ok, the last integral we can calculate directly

i will edit my answer it will be easier to see

JacobsonRadical

okay.

thanks!

infinity

it will take one more minute, the equations are long

JacobsonRadical

no problem

take your time

I just figured it out we have $\sqrt{1-\cos y}=\sqrt{2}\sqrt{\sin^{2}(y/2)}$

but please edit so that people in the future know what to do.

Thanks!

infinity

Oh wait, your way can be simpler than mine@JacobsonRadical

look at my answer :P

JacobsonRadical

07:51

see it!

perfect!

I also got $8$

now we are sure that you are correct!!

infinity

great :)

thanks for the correction..

JacobsonRadical

thank you so much for helping me in the late night.

infinity

no problem

JacobsonRadical

no problem. thanks for writing them out.

Have a good evening :)

infinity

do you study math?

JacobsonRadical

07:52

yes

master in math

:( try to get a PhD but hard

infinity

nice! me too

im doing masters too

where do you study?

JacobsonRadical

oh!

NYU Courant

infinity

cool

JacobsonRadical

where are you?

(Well Master in Courant is really a torture... its like in the middle. I took PhD course so no one likes me and the professor does not like everybody....)

infinity

im studying at HUJI

@JacobsonRadical what course is that?

JacobsonRadical

08:02

this is Harmonic analysis

I am arguing some $f_{k}(x):=f(kx)$ has no $L^{1}$ convergence limit

and what I did in the post is a counterexample

you are in Jerusalem!!??

wow

cool

infinity

@JacobsonRadical Yep :)

@JacobsonRadical wait what do you mean?

you showed that $e^{ix}$ is not the limit

JacobsonRadical

yes

the question gives a general $f\in L^{1}(\mathbb{S}^{1})$

and define $f_{k}(x):=f(kx)$

and asked does $f_{k}(x)$ have a limit in $L^{1}$

well so I said $f_{k}(x)$ does not necessarily have such a limit

due to this counter example

$f(x):=e^{ix}$ and $f_{k}(x)=e^{ikx}$

infinity

08:21

Ah. got it..

JacobsonRadical

yeah..

so we have time difference right? you are still morning or afternoon?

infinity

morning , here its 10:30 am

JacobsonRadical

wow

I am 3:29 am

infinity

wow why are you still awake ^^

JacobsonRadical

my fiancee is working overnight, she has a heavy task

i just be with her

by working on my Homework :(

infinity

08:32

well then i guess you both working overnight

JacobsonRadical

ahah yes

I don't know. I tried to live a healthier life

but it is just not me anymore

I need to stay up late hahahaha

infinity

lol

so you sleep during the day ?

JacobsonRadical

yeah

bad habit from my undergraduate years

hard to modify..

infinity

yeah , i also studied at night in my undergraduate years, but not at 3:30 ^^ maximum was like 1-2

JacobsonRadical

hahahah

it seems like undergraduates are the same all over the world

infinity

08:55

yep

JacobsonRadical

need to sleep. Have a wonderful day!

my fiancee finishes her task, good

infinity

good to hear!

good night / morning :P

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Mar '206

Mar21

$Mathematics$ Discussion between infinity and Jacob…

Imported from a comment discussion on math.stackexchange.com/q...

analysis complex-analysis complex-integration lebesgue-integral

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