Calculus and analysis

Martin Sleziak

08:27

0

Q: If $f$ belongs to $C[0, 1]$. Determine the cases where the given condition implies that $f$ is identically zero

If $f$ belongs to $C[0, 1]$. Determine the cases where the given condition implies that $f$ is identically zero. Case 1 $\int\ x^nf(x) =0$ for all non negative n Case 2 $\int\ f(x)Cosnx =0$ for all non negative n Case 3 $\int\ f(x)Sinnx =0$ for all positive n What I know is ...

@Haha Can u help me in this question? If $f$ belongs to $C[0, 1]$. Determine the cases where the given condition implies that $f$ is identically zero. Case 1 $\int\ x^nf(x) =0$ for all non negative n Case 2 $\int\ f(x)Cosnx =0$ for all non negative n Case 3 $\int\ f(x)Sinnx =0$ for all positive n — sani 7 hours ago

@sani Asking random questions in comments is not a reasonable way how to deal with the question ban. (Neither is asking questions on other sites, where they are off-topic.) You have enough reputation to talk in chat, so you can try to ask there. For example Calculus and analysis chat room or the main chat room might be a reasonable place for the question from your comment. — Martin Sleziak 7 secs ago

BTW it is very likely that the above questions have been already asked on the main.

For example, after a quick search using Approach0 I found these:

5

Q: If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$.

Possible Duplicates: Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$ Slight generalization of an exercise in (blue) Rudin What can we say about $f$ if $\int_0^1 f(x)p(x)dx=0$ for all polynomials $p$? I found a nice problem I would like to share. Pro...

real-analysis

12

Q: What can we say about $f$ if $\int_0^1 f(x)p(x)dx=0$ for all polynomials $p$?

This question was motivated by another question in this site. As explained in that problem (and its answers), if $\displaystyle f$ is continuous on $\displaystyle [0,1]$ and $\displaystyle \int_0^1 f(x)p(x)dx=0$ for all polynomials $\displaystyle p$, then $\displaystyle f$ is zero everywhere. S...

real-analysis

7

Q: Slight generalization of an exercise in (blue) Rudin

Exercise 7.20 of blue Rudin (Principles of Mathematical Analysis), 3rd edition, says: If $f$ is continuous on $[0,1]$ and if $$\int_0^1f(x)x^n\,dx = 0, (n=0,1,2,\ldots),$$ prove that $f(x)=0$ on $[0,1].$ One proof: Let $\{p_n\}$ be a sequence of polynomials uniformly approximating $f.$ Then $...

real-analysis

19

Q: Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$

As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem proving that if (on $C([0, 1])$) $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 0$, then $f$ must be ident...

real-analysis analysis integration

In fact here is a rather similar question to yours:

1

Q: Let $f : [0;1] \to \mathbb{R}$ be a continuous function such that $f(0) = 0$. Which of the following statements are true?

Let $f : [0;1] \to \mathbb{R}$ be a continuous function such that $f(0) = 0$. Which of the following statements are true? a. If $\int_ 0^{\pi} f(t) \cos nt\, dt = 0,$ for all $n \in {0} \cup \mathbb{N}$, then $f= 0$. b. If $\int_ 0^{\pi} f(t) \sin nt\, dt = 0,$ for all $n \in \mathbb{N}$, then $...

real-analysis fourier-series

Martin Sleziak

09:11

But this is not a random question...This is very much relevant to the main question..However If u know the answer U could have helped me with lesser effort..@MartinSleziak — sani 7 mins ago

sani

$\int_{0}^{\pi}f(x)\sin nx =0$ for all positive integer n then prove $f(x)$ is identically $0$ on $[0, \pi]$ where $f(x)$ is a continuous function on $[0, 1]$ Can anyone plzzzzzzzzzzzzzzz help me...I got stuck at this problem...I would have asked this question on the forum..But I am blocked.. Plzzz help me...Thanks in Advance..

Martin Sleziak

@sani So is it continuous on $[0,\pi]$ or on $[0,1]$?

Doesn't the question linked above answer your question?

Just in case - since you are probably new to chat: For instructions how to render MathJax(TeX) in chat see this post on meta. Or directly robjohn's website.

sani

@MartinSleziak sorry sir [0, \pi]

$[0, \pi]$

Okk Sir I will see...Now can u plzz give any hint

no sir that was a different question..$f(0) = 0$ is given

Martin Sleziak

So what is actually the question? We have $\int_0^\pi f(x) \sin nx dx=0$ for $f$ continuous on $[0,\pi]$?

And what are you actually allowed to use. For example, can you use Fourier series?

Here is one more similar question:

10

Q: If $ \int_0^{\pi}f(x)\cos(nx)\,dx=0$ for all $n$ then prove that $f\equiv 0$

If $f:[0,\pi]\to \mathbb R$ is continuous and $f(0)=0$ such that $\displaystyle \int_0^{\pi}f(x)\cos(nx)\,dx=0$ for all $n=0,1,2,\cdots$ then prove that $f\equiv 0$ in $[0,\pi]$. I want to apply Weierstrass approximation theorem. As $f$ is continuous so there exists a sequence of polynomials...

real-analysis integration analysis definite-integrals

4

Q: Application of Fourier Series and Stone Weierstrass Approximation Theorem

If $f \in C[0, \pi]$ and $\int_0^\pi f(x) \cos nx\, \text{d}x = 0$ , then $f = 0$ Define $ g(x) = \begin{cases} f(-x) & \text{if } -\pi \leq x < 0;\\ f(x) & \text{if } 0 \leq x \leq \pi. \end{cases}$ which is an even function So $g(x)$ can be written as $\sum_{n=0...

real-analysis proof-verification continuity fourier-series

Both from this search.

Reading the answers to the above questions, it seems not very likely that there is an easy answer without at least some result on Fourier series.

It seems that you're gone. I guess the above links could help you.

Martin Sleziak

11:24

hiii@Martin Sleziak Do u have any Idea How can I proceed????? — sani 42 mins ago

@sani I am not sure how you are using pings, but I did not receive any notification on your comments.

I can hardly give you any advice, since you did not really clarified what is actually the question.

2 hours ago, by Martin Sleziak

So what is actually the question? We have $\int_0^\pi f(x) \sin nx dx=0$ for $f$ continuous on $[0,\pi]$?

2 hours ago, by Martin Sleziak

And what are you actually allowed to use. For example, can you use Fourier series?

2 hours ago, by Martin Sleziak

Reading the answers to the above questions, it seems not very likely that there is an easy answer without at least some result on Fourier series.

I guess that you have leave a space before @ for a notification to work. (This would explain why I was not notified by either of your comments.)

sani

11:50

@MartinSleziak Sir I am really sorry..Actually I am new here.. That's why I could not operate corretcly..U can use anything u want...Plzzz mention it clearly only...And I have edited the question clearly....U plzzz go through the question rigorously once..

Martin Sleziak

@sani Which question are you actually talking about?

Could you give me a link, since you are saying that you edited the question?

sani

Okk sIR I am giving u the question..Plzz wait a minute..

Martin Sleziak

And please do not call me sir. That's really upsetting.

sani

If $f$ belongs to $C[0, \pi]$. Determine the cases where the given condition implies that $f$ is identically zero.

Case 1 $\int_{0}^{\pi}\ x^nf(x) =0$ for all non negative integer n

Case 2 $\int_{0}^{\pi}f(x)\cos nx =0$ for all non negative integer n

Case 3 $\int_{0}^{\pi}f(x)\sin nx =0$ for all positive integer n

If $f$ belongs to $C[0, \pi]$. Determine the cases where the given condition implies that $f$ is identically zero.

Case 1     $\int_{0}^{\pi}\ x^nf(x) =0$     for all non negative integer n

Case 2    $\int_{0}^{\pi}f(x)\cos nx =0$    for all non negative integer n

Case 3  $\int_{0}^{\pi}f(x)\sin nx =0$    for all positive integer n

Martin Sleziak

Let's have a look at the first one.

@sani As I have mentioned above, there are several posts on the main where you can find answers for this one: If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$., Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$, etc.

sani

11:58

Yea First case seem easy..It can be done by Weirstrass

seems*

Martin Sleziak

For the second one, I have also given you some links, like this one: If $ \int_0^{\pi}f(x)\cos(nx)\,dx=0$ for all $n$ then prove that $f\equiv 0$

Answers in the links I have found use Fourier series in some way.

sani

Thank UUUU...I am trying to understand...and what about case 4..Will it be same as 3?

Martin Sleziak

There is no case 4...?

Do you mean case 3? (The one with $\sin nx$?)

sani

yes yes case 3

Martin Sleziak

I'd guess that it can probably be solved similarly. Now you want to have Fourieser series only with sines, so you probably want to extend $f$ to an odd function on $[-\pi,\pi]$.

sani

12:05

Okkk I am trying to solve..I will let u know once I am done...

Martin Sleziak

As I said before, it is basically this question: Let $f : [0;1] \to \mathbb{R}$ be a continuous function such that $f(0) = 0$. Which of the following statements are true?

Martin Sleziak

12:29

The title says $[0,1]$, but the question actually asks about integrals from $0$ to $\pi$.

@BAYMAX If you are around - IIRC you managed to somehow get out of question ban. Maybe you will have some reasonable advice for sani.

Sir Actually I am blocked. I can not ask question. Can u plzz help me?@User8128 — sani May 19 at 1:19

sani

12:43

I did not get u

sani

13:04

$$\hat f (n) = \frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}\,dx = \frac{1}{\pi}\int_{0}^\pi f(x)\cos (nx)\,dx = 0$$

Can u plzz tell me how this happens???@Martin Sleziak

Martin Sleziak

13:14

@sani Isn't that directly the definition of Fourier coefficients?

$$b_n = \frac{1}{\pi}\int_{-\pi}^{\pi}s(x) \sin(nx)\, dx$$

$$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}s(x) \cos (nx)\, dx$$

sani

no no that's okay.. I haVe problem with this equality..How R.H.S = L.H.S

Martin Sleziak

I am not really sure what you now mean by RHS and LHS.

Is this the problem? $$e^{-inx} = \cos nx + i \sin nx$$

You can basically ignore that part. If you are used to write Fourier series in the form $$f \sim a_0 + \sum_{n=1}^\infty a_n\sin nx + b_n\cos nx$$ then simply use this form.

sani

Can u PLZZ explain how Parseval's identity is working here for case 2 and 3..Everyone is talking about this...

Martin Sleziak

Well, I doubt I can say something which you do not find already in Wikipedia article: Parseval's identity.

Here you show in both cases that all coefficients are zeroes. (And then apply the identity.)

sani

Can u plzzz elaborate???? In one place this equality $$\hat f (n) = \frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}\,dx = \frac{1}{\pi}\int_{0}^\pi f(x)\cos (nx)\,dx = 0$$ is used..I have problem here...When f(x) is even then how it happens?

Martin Sleziak

13:29

If $f$ is even then $g(x)=f(x)\sin nx$ is odd and $h(x)=f(x)\cos nx$ is even.

sani

math.stackexchange.com/questions/1514005/… see this link

Martin Sleziak

For odd function $g(x)$ you have
$$\int_{-\pi}^\pi g(x) dx = 0.$$

For even function $h(x)$ you have
$$\int_{-\pi}^\pi h(x) dx = 2\int_{0}^\pi h(x) dx .$$

sani

ohhh I get it

thank u once again..

Actually My brain stopped working... I have disturbed u enough .. Sorry for that...

BAYMAX

16:23

Yes @MartinSleziak

So @sani try to answer the questions in the main correctly which you think you can answer well so that your answers get up voted and you get more points or reputation and you see that you can ask questions now,my reputation was 96 that time during the block and I incresed upto some 300 or 400 and I saw that the block of ask question was lifted for me! so try to answer as much questions as you can correctly with proper LaTeX and clearly . ok all the best!

sani

17:58

@BAYMAX Thank u for ur wishes.. I will tryy....

Transcript for

$Mathematics$ Calculus and analysis