Does Bessel's Inequality Imply Fourier Coefficients Approach Zero?

[Joystar77]

1. Let c > 0 be a constant. For F E Cp (-C, C), the Bessel inequality is

ao squared/ 2 + N E n = 1 (an squared + bn squared) is less than or equal to 1/C { c on top, c on bottom [ f (x)] squared of x, N = 1, 2, ...

Where an and bn are the Fourier coefficients for f (x), -c < x < c.

Using this inequality show that lim an = 0, n ---> infinity and lim bn = 0, n ----> infinity.

Is this the correct way of working out this problem?

 

[Joystar77]

Sorry, I forgot to mention that I am totally lost and don't know where to start with this problem.

 

[M R]

Hi Joystar1977, I think I might be able to help with some of your recent questions but I'm not quite sure what the questions are exactly. If you look at the pdf linked to http://mathhelpboards.com/latex-tips-tutorials-56/math-help-boards-latex-guide-pdf-1142.html you can quickly learn enough LaTeX to write your maths in a much more readable form.

E.g. \(\displaystyle \sum_{r=1}^n r=\frac{1}{2}n(n+1)\)

 

[Joystar77]

I hope that this problem above is easier to read. Here it is again as follows:

Let c > o be a constant. For F € Cp, the Bessel inequality is

〖^2〗a_0/2+∑_(n=1)^N▒(a_n 〖^2〗+b_n 〖^2〗≤1/c[c on top,-c on bottom {f (x)}〖〖^2〗 of x,N=1,2,…〗⁡ )
Where an and bn are the Fourier coefficients for f (x), -c < x < c

Using this inequality show that

〖lim⁡an=0,〗┬(n→∞)⁡〖lim⁡bn=0〗┬(n→∞)⁡

 

[Evgeny.Makarov]

I hope that this problem above is easier to read.

To be honest, it's not much easier. Do you mean the following?

Let $c > 0$ be a constant. For $f\in C_p(-c,c)$, the Bessel inequality is \[\frac{a_0^2}{2} +\sum_{k=1}^N (a_k^2+b_k^2) \le\frac{1}{c} \int_{-c}^c(f(x))^2\,dx,\quad N=1,2,\dots\tag{*}\] where $a_n$ and $b_n$ are the Fourier coefficients for $f(x)$, $-c < x < c$.

Using this inequality show that \[\lim\limits_{n\to\infty} a_n=0,\quad \lim_{n\to\infty}⁡b_n=0\enspace.\]

Does $C_p(-c,c)$ here means the set of piecewise continuous functions?

I am not sure what you have covered with respect to series and sequence convergence, but the key here is the $n$th term test. Since the series in the left-hand side of (*) converges, $a_n^2+b_n^2\to0$ as $n\to\infty$.

A couple of administrative issues. Please read the link provided in post #3 about LaTeX on this forum. If you click "Reply With Quote" button under this post, you can use the LaTeX code in it as an example. Using LaTeX is crucial for questions like this, with complicated formulas. You could also post a photo of a question. Second, rule #5 from the http://mathhelpboards.com/rules/ asks one to choose a subforum not based on the course from which the question has come, but on the content of the question itself. Since this question is about convergence, it should be posted in the Calculus subforum.

 

[Joystar77]

Yes, Evgeny! Accept when using the Math Math symbol on top of it is a capital N and the bottom saying n=1 (instead of k=1). Also, instead of a of k squared and b of k squared. It is suppose to be a of n squared and b of n squared. Also, I wasn't too sure which section and figured that since I am in a Discrete Mathematics course then it would fall under the Discrete Mathematics category. Thanks for teaching me something new! I know for next time if a math problem has to do with Convergence, then it goes under the Calculus category.

To be honest, it's not much easier. Do you mean the following?

Let $c > 0$ be a constant. For $f\in C_p(-c,c)$, the Bessel inequality is \[\frac{a_0^2}{2} +\sum_{k=1}^N (a_k^2+b_k^2) \le\frac{1}{c} \int_{-c}^c(f(x))^2\,dx,\quad N=1,2,\dots\tag{*}\] where $a_n$ and $b_n$ are the Fourier coefficients for $f(x)$, $-c < x < c$.

Using this inequality show that \[\lim\limits_{n\to\infty} a_n=0,\quad \lim_{n\to\infty}⁡b_n=0\enspace.\]

Does $C_p(-c,c)$ here means the set of piecewise continuous functions?

I am not sure what you have covered with respect to series and sequence convergence, but the key here is the $n$th term test. Since the series in the left-hand side of (*) converges, $a_n^2+b_n^2\to0$ as $n\to\infty$.

A couple of administrative issues. Please read the link provided in post #3 about LaTeX on this forum. If you click "Reply With Quote" button under this post, you can use the LaTeX code in it as an example. Using LaTeX is crucial for questions like this, with complicated formulas. You could also post a photo of a question. Second, rule #5 from the http://mathhelpboards.com/rules/ asks one to choose a subforum not based on the course from which the question has come, but on the content of the question itself. Since this question is about convergence, it should be posted in the Calculus subforum.

 

[Joystar77]

Evgeny: I forgot to answer your other question. Can I ask you something please? How am I suppose to know about a math problem for the content itself? I figured that since I was in a Discrete Mathematics course that the questions should strictly relate to Discrete Mathematics. I will be honest that I am not familiar on how to base the content itself when it comes to a math question. Can you give me tips or clues on what to look for in math problems?

To be honest, it's not much easier. Do you mean the following?

Let $c > 0$ be a constant. For $f\in C_p(-c,c)$, the Bessel inequality is \[\frac{a_0^2}{2} +\sum_{k=1}^N (a_k^2+b_k^2) \le\frac{1}{c} \int_{-c}^c(f(x))^2\,dx,\quad N=1,2,\dots\tag{*}\] where $a_n$ and $b_n$ are the Fourier coefficients for $f(x)$, $-c < x < c$.

Using this inequality show that \[\lim\limits_{n\to\infty} a_n=0,\quad \lim_{n\to\infty}⁡b_n=0\enspace.\]

Does $C_p(-c,c)$ here means the set of piecewise continuous functions?

I am not sure what you have covered with respect to series and sequence convergence, but the key here is the $n$th term test. Since the series in the left-hand side of (*) converges, $a_n^2+b_n^2\to0$ as $n\to\infty$.

A couple of administrative issues. Please read the link provided in post #3 about LaTeX on this forum. If you click "Reply With Quote" button under this post, you can use the LaTeX code in it as an example. Using LaTeX is crucial for questions like this, with complicated formulas. You could also post a photo of a question. Second, rule #5 from the http://mathhelpboards.com/rules/ asks one to choose a subforum not based on the course from which the question has come, but on the content of the question itself. Since this question is about convergence, it should be posted in the Calculus subforum.

 

[Evgeny.Makarov]

I figured that since I was in a Discrete Mathematics course that the questions should strictly relate to Discrete Mathematics.

I would be surprised to see this question in a Discrete Math course. Discrete math usually does not deal with such topics as continuous functions, limits and integrals. In fact, introductory discrete math courses do not usually (but not always) deal with real numbers.

I will be honest that I am not familiar on how to base the content itself when it comes to a math question. Can you give me tips or clues on what to look for in math problems?

If you know that continuity, limits and integrals are topics from calculus, then you know where to put such question. Otherwise, you may have doubts that this question fits into discrete math since it talks about continuous functions, and continuous is the opposite of discrete. Then click the triangle with the exclamation point below and to the left of the post and report it to the moderators, asking for an advice where to put the question. This is described in the same rule #5; please read it!

Do you have more questions about this problem (the math one, not the administrative issue)?

 

[Joystar77]

Alright, then I would just look at the topics for mathematics and the sub topics of what is being talked about in the math problem. If something doesn't exactly state the mathematical term, then possibly look at other words that have the same meaning such as Integral:
a function of which a given function is the derivative, i.e., which yields that function when differentiated, and which may express the area under the curve of a graph of the function; a function satisfying a given differential equation. Am I correct to possibly look for another word that has the same meaning in mathematics?

I would be surprised to see this question in a Discrete Math course. Discrete math usually does not deal with such topics as continuous functions, limits and integrals. In fact, introductory discrete math courses do not usually (but not always) deal with real numbers.

If you know that continuity, limits and integrals are topics from calculus, then you know where to put such question. Otherwise, you may have doubts that this question fits into discrete math since it talks about continuous functions, and continuous is the opposite of discrete. Then click the triangle with the exclamation point below and to the left of the post and report it to the moderators, asking for an advice where to put the question. This is described in the same rule #5; please read it!

Do you have more questions about this problem (the math one, not the administrative issue)?

 

[Evgeny.Makarov]

Look, don't worry about nomenclature. Spend your time studying theory and then solving problems. If you invest enough time in this, you'll become better.

 

[Joystar77]

Evgeny: In response to your question, yes Cp (-c, c) does mean the set of piecewise continuous functions.