Page 1 of 2 12 LastLast
Results 1 to 15 of 23

Math Help - Fourier transform and orthogonal system

  1. #1
    Member
    Joined
    Sep 2010
    From
    Germany
    Posts
    109
    Thanks
    4

    Fourier transform and orthogonal system

    Let \phi_{n}(x)=\begin{cases} 1 ,\ x\in[n-\frac{1}{2},n+\frac{1}{2}]\\ 0, \ elsewhere \end{cases}

    a) Compute the Fourier transform of the function
    (\hat{\phi_{n}})(\xi)=\int_{n-\frac{1}{2}}^{n+\frac{1}{2}}e^{-i\xi x}dx=-\frac{1}{i\xi}\Bigl[e^{-i\xi x}\Bigr]_{n-\frac{1}{2}}^{n+\frac{1}{2}}=-\frac{1}{i\xi}\Bigl[e^{-i\xi n}e^{-i\xi\frac{1}{2}}-e^{-i\xi n}e^{i\xi\frac{1}{2}}\Bigr]=\frac{1}{\xi i}\Bigl[-e^{-i\xi n}e^{-i\xi\frac{1}{2}}+e^{-i\xi n}e^{i\xi\frac{1}{2}}\Bigr]=\frac{2e^{-i\xi n}}{\xi}\Bigl[\frac{e^{i\xi \frac{1}{2}}-e^{-i\xi \frac{1}{2}}}{2i}\Bigr]=\frac{2e^{-i\xi n}}{\xi}\sin(\frac{\xi}{2})

    is this part correct?

    b) prove that (\hat{\phi_{n}})_{n=1}^{\infty} is an orthogonal system in  L^2(\mathbb{R}). Prove that it is not complete.

    Could someone show me how to do the part b?
    thank you very much
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: Fourier transform and orthogonal system

    Quote Originally Posted by rayman View Post
    Let \phi_{n}(x)=\begin{cases} 1 ,\ x\in[n-\frac{1}{2},n+\frac{1}{2}]\\ 0, \ elsewhere \end{cases}

    a) Compute the Fourier transform of the function
    (\hat{\phi_{n}})(\xi)=\int_{n-\frac{1}{2}}^{n+\frac{1}{2}}e^{-i\xi x}dx=-\frac{1}{i\xi}\Bigl[e^{-i\xi x}\Bigr]_{n-\frac{1}{2}}^{n+\frac{1}{2}}=-\frac{1}{i\xi}\Bigl[e^{-i\xi n}e^{-i\xi\frac{1}{2}}-e^{-i\xi n}e^{i\xi\frac{1}{2}}\Bigr]=\frac{1}{\xi i}\Bigl[-e^{-i\xi n}e^{-i\xi\frac{1}{2}}+e^{-i\xi n}e^{i\xi\frac{1}{2}}\Bigr]=\frac{2e^{-i\xi n}}{\xi}\Bigl[\frac{e^{i\xi \frac{1}{2}}-e^{-i\xi \frac{1}{2}}}{2i}\Bigr]=\frac{2e^{-i\xi n}}{\xi}\sin(\frac{\xi}{2})

    is this part correct?

    b) prove that (\hat{\phi_{n}})_{n=1}^{\infty} is an orthogonal system in  L^2(\mathbb{R}). Prove that it is not complete.

    Could someone show me how to do the part b?
    thank you very much
    (a) looks OK

    (b) Start by showing that \langle \widehat{\phi}_n, \widehat{\phi}_m\rangle=0 when n\ne m.

    To show that it is not compelete I think you need to find a function in  L^2(\mathbb{R}) that cannot be represented as a series in the \widehat{\phi}_n convergent in norm.

    CB
    Last edited by CaptainBlack; December 13th 2011 at 02:36 AM. Reason: LaTeX typpo that made post nonsense (Hats did not render:( bighat <> widehat)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7

    Re: Fourier transform and orthogonal system

    For (b), I would start by taking f to be a function in  L^2(\mathbb{R}) that is orthogonal to all the functions \phi_n. For example, f(x) = \begin{cases} x & x\in[-\frac{1}{2},\frac{1}{2}],\\ 0 & \text{elsewhere.} \end{cases}

    Parseval's theorem then tells you that \hat{f} is orthogonal to all the functions \hat{\phi}_n.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Sep 2010
    From
    Germany
    Posts
    109
    Thanks
    4

    Re: Fourier transform and orthogonal system

    I am not sure If I understand what I am supposed to do in part b)
    I thougt than to prove the system to be orthogonal we need to compute the inner product of our transformed function \hat{(\phi_{n})} with some other function and if
     \Bigl<\hat{(\phi_{n})},\hat{(\phi_{m})}\Bigr>and control if this equals to 1 if m=n or 0 otherwise. Then we prove that \hat{(\phi_{n})} is an orthogonal system right? but how do we choose /find the other function \hat{(\phi_{m})}??
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7

    Re: Fourier transform and orthogonal system

    Quote Originally Posted by rayman View Post
    I am not sure If I understand what I am supposed to do in part b)
    I thougt than to prove the system to be orthogonal we need to compute the inner product of our transformed function \hat{(\phi_{n})} with some other function and if
     \Bigl<\hat{(\phi_{n})},\hat{(\phi_{m})}\Bigr>and control if this equals to 1 if m=n or 0 otherwise. Then we prove that \hat{(\phi_{n})} is an orthogonal system right? but how do we choose /find the other function \hat{(\phi_{m})}??
    To find \hat{\phi}_m, you just replace n by m in the formula that you obtained for \hat{\phi}_n. But to compute \bigl\langle\hat{\phi}_n,\hat{\phi}_m\bigr\rangle, you should use Parseval's theorem, which tells you that \bigl\langle\hat{\phi}_n,\hat{\phi}_m\bigr\rangle = \bigl\langle\phi_n,\phi_m\bigr\rangle. That is much easier to calculate than the inner product of the Fourier transforms, and it will show you that the system \hat{(\phi_{n})} is orthonormal. To show that it is not a complete system, you must find a nonzero L^2-function that is orthogonal to each \hat{\phi}_n. The way to do that is again by using Parseval's theorem, as indicated in my previous comment.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: Fourier transform and orthogonal system

    Quote Originally Posted by CaptainBlack View Post
    (a) looks OK

    (b) Start by showing that \langle \widehat{\phi}_n, \widehat{\phi}_m\rangle=0 when n\ne m.

    To show that it is not compelete I think you need to find a function in  L^2(\mathbb{R}) that cannot be represented as a series in the \widehat{\phi}_n convergent in norm.

    CB




    It is easier to find a counter example than you might think, since almost any element of L^2(\mathbb{R}) that decays faster than 1/\xi cannot be represented. (the space spanned by this set of functions is a subset of a set of periodic function of period 2\pi times a \rm {sinc} function - a bit of care and "set" can be replaced by "space" but I'm not minded to be careful today)
    Last edited by CaptainBlack; December 13th 2011 at 07:28 PM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Aug 2010
    Posts
    886
    Thanks
    91

    Re: Fourier transform and orthogonal system

    b) Incomplete because you can't represent an arbitrary function with basis functions that are 0 for all x < x0.

    Like trying to represent an arbitrary vector in 3d with basis vectors that have 0 as the first component.
    Last edited by Hartlw; December 13th 2011 at 12:06 PM. Reason: add "basis" to functions
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: Fourier transform and orthogonal system

    Quote Originally Posted by Hartlw View Post
    b) Incomplete because you can't represent an arbitrary function with basis functions that are 0 for all x < x0.

    Like trying to represent an arbitrary vector in 3d with basis vectors that have 0 as the first component.
    I thought I had already posted this but ...

    The basis functions are not all zero beyoud some point, they are a sinc function times the elements of the standard fourier series basis \exp(-{\rm{i}} n \xi), \ \ n= ... -1,0,1, ....
    CB
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Super Member
    Joined
    Aug 2010
    Posts
    886
    Thanks
    91

    Re: Fourier transform and orthogonal system

    Quote Originally Posted by CaptainBlack View Post
    I thought I had already posted this but ...

    The basis functions are not all zero beyoud some point, they are a sinc function times the elements of the standard fourier series basis \exp(-{\rm{i}} n \xi), \ \ n= ... -1,0,1, ....
    CB
    You can't create a complete basis using only an incomplete basis.


    Edit: While I was responding on my computer the screen went blank and locked. After I shut down and restarted I could access the forum home pg but nothing else. Went to another computer (this one, with later version of Windows) and managed to respond, but when I shut down after having done nothing else I got notice of down-loading fifteen updates. That's usually a sign of a detected attack. When I rebooted there was a long list of registry updates. Scary. Concidence? Censorship?

    Edit Update: Now can access MHF from my computer. Looks like whatever happened was unrelated to Forum, but detected by security software and corrected. Whew!
    Last edited by Hartlw; December 14th 2011 at 09:34 AM.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Super Member
    Joined
    Aug 2010
    Posts
    886
    Thanks
    91

    Re: Fourier transform and orthogonal system

    This is awkward but I simply can't get TEX to do this. So

    xn = phi sub n
    x'n = hat phi sub n

    You can't create a complete basis x'n using only an incomplete basis xn. xn was previously shown to be incomplete.

    There is also a direct test of x'n which might work out:

    "An orthonormal sequence in H is complete iff the condition (x,x'n) = 0 for all n implies x = 0."

    You could write out the explicit expression for (x,x'n) and see if there is an obvious non-zero x which satrisfies the condition.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: Fourier transform and orthogonal system

    Quote Originally Posted by Hartlw View Post
    This is awkward but I simply can't get TEX to do this. So

    xn = phi sub n
    x'n = hat phi sub n

    You can't create a complete basis x'n using only an incomplete basis xn. xn was previously shown to be incomplete.
    Where? Not in this thread, you are appealing to a result that we do not know that you know, though you could use the properties of the FT to justify this on the fly.

    CB
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Super Member
    Joined
    Aug 2010
    Posts
    886
    Thanks
    91

    Re: Fourier transform and orthogonal system

    Quote Originally Posted by Hartlw View Post
    b) Incomplete because you can't represent an arbitrary function with basis functions that are 0 for all x < x0.

    Like trying to represent an arbitrary vector in 3d with basis vectors that have 0 as the first component.
    That's what I originally wrote in post 7. I showed \phi_n was incomplete. But that's ok, because if \phi_n is incomplete, \widehat{\phi}_n is incomplete because you can't create a complete basis using only an incomplete basis.

    There is another way to show \phi_n is incomplete using theorem from my previous post. Show (f, \phi_n) = 0 for a non-zero f. take f(x) = 0 for x>0 and f(x) =1 for a<= x <=0, any a.

    NOTE: finally figured out how to write \phi_n. I copied the code and tried it in a practice post and wrapped it in [tex] tags and it simply did not work. Then tried [TEX] tags and it worked. Now I can communicate a little more intelligibly.

    BY the way, when I tried to reply on my other computer, Forum locked up on me again. I noticed a current thread about hacking. Maybe something is around.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: Fourier transform and orthogonal system

    Quote Originally Posted by Hartlw View Post
    b) Incomplete because you can't represent an arbitrary function with basis functions that are 0 for all x < x0.



    Like trying to represent an arbitrary vector in 3d with basis vectors that have 0 as the first component.
    But the \phi_n are not all zeros for all x less than some x_0, or rather you seem to be assuming that n \in \mathbb{N} rather than n \in \mathbb{Z} and there seems to be no reason to assume so (It was rather sloppy of the OP not to specify what the index set was/is).

    Also, it is sloppy not to specify which basis you are talking about, I originally assumed you were talking about \{\widehat{\phi}_n, n\in \mathbb{Z}\}, rather than \{{\phi}_n, n\in \mathbb{Z}\}. It is also sloppy to rely on results in other threads without explicitly referencing them and giving a link to the thread (which I pressume was your intention from some of what you have written in other posts in this thread).



    That's what I originally wrote in post 7. I showed \phi_n was incomplete. But that's ok, because if \phi_n is incomplete, \widehat{\phi}_n is incomplete because you can't create a complete basis using only an incomplete basis.



    There is another way to show \phi_n is incomplete using theorem from my previous post. Show (f, \phi_n) = 0 for a non-zero f. take f(x) = 0 for x>0 and f(x) =1 for a<= x <=0, any a.



    NOTE: finally figured out how to write \phi_n. I copied the code and tried it in a practice post and wrapped it in [tex] tags and it simply did not work. Then tried [TEX] tags and it worked. Now I can communicate a little more intelligibly.



    BY the way, when I tried to reply on my other computer, Forum locked up on me again. I noticed a current thread about hacking. Maybe something is around.
    The easiest way to do this is to show that \chi_{{[-1/4,1/4]}} cannot be represented in the basis \{\phi_n, n\in \mathbb{Z}\}, and so \mathfrak{F} \chi_{[-1/4,1/4]} cannot be represented in the basis \{\widehat{\phi}_n, n\in \mathbb{Z}\}

    CB
    Last edited by CaptainBlack; December 14th 2011 at 10:56 PM.
    Follow Math Help Forum on Facebook and Google+

  14. #14
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7

    Re: Fourier transform and orthogonal system

    Quote Originally Posted by CaptainBlack View Post
    Quote Originally Posted by Hartlw View Post
    This is awkward but I simply can't get TEX to do this. So

    xn = phi sub n
    x'n = hat phi sub n

    You can't create a complete basis x'n using only an incomplete basis xn. xn was previously shown to be incomplete.
    Where? Not in this thread
    Has my comment #3 become invisible then?
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: Fourier transform and orthogonal system

    Quote Originally Posted by Opalg View Post
    Has my comment #3 become invisible then?
    Your f is \phi_0 under my assumption about the unspecified index set for the \phi s (but the basic idea can be made to work)

    CB
    Follow Math Help Forum on Facebook and Google+

Page 1 of 2 12 LastLast

Similar Math Help Forum Discussions

  1. fourier transform of sin(3w)*cos(w)/(w^2)
    Posted in the Calculus Forum
    Replies: 7
    Last Post: July 2nd 2011, 06:03 PM
  2. Laplace transform and Fourier transform what is the different?
    Posted in the Advanced Applied Math Forum
    Replies: 8
    Last Post: December 29th 2010, 10:51 PM
  3. [SOLVED] Fourier Transform of x^2*sin(x)
    Posted in the Differential Equations Forum
    Replies: 25
    Last Post: July 22nd 2010, 03:42 PM
  4. Replies: 0
    Last Post: April 23rd 2009, 05:44 AM
  5. from fourier transform to fourier series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 1st 2008, 06:35 AM

Search Tags


/mathhelpforum @mathhelpforum