Fourier transform and orthogonal system

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

a) Compute the Fourier transform of the function

$\displaystyle (\hat{\phi_{n}})(\xi)=\int_{n-\frac{1}{2}}^{n+\frac{1}{2}}e^{-i\xi x}dx=-\frac{1}{i\xi}\Bigl[e^{-i\xi x}\Bigr]_{n-\frac{1}{2}}^{n+\frac{1}{2}}=-\frac{1}{i\xi}\Bigl[e^{-i\xi n}e^{-i\xi\frac{1}{2}}-e^{-i\xi n}e^{i\xi\frac{1}{2}}\Bigr]=\frac{1}{\xi i}\Bigl[-e^{-i\xi n}e^{-i\xi\frac{1}{2}}+e^{-i\xi n}e^{i\xi\frac{1}{2}}\Bigr]=\frac{2e^{-i\xi n}}{\xi}\Bigl[\frac{e^{i\xi \frac{1}{2}}-e^{-i\xi \frac{1}{2}}}{2i}\Bigr]=\frac{2e^{-i\xi n}}{\xi}\sin(\frac{\xi}{2})$

is this part correct?

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

Could someone show me how to do the part b?

thank you very much

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

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

a) Compute the Fourier transform of the function

$\displaystyle (\hat{\phi_{n}})(\xi)=\int_{n-\frac{1}{2}}^{n+\frac{1}{2}}e^{-i\xi x}dx=-\frac{1}{i\xi}\Bigl[e^{-i\xi x}\Bigr]_{n-\frac{1}{2}}^{n+\frac{1}{2}}=-\frac{1}{i\xi}\Bigl[e^{-i\xi n}e^{-i\xi\frac{1}{2}}-e^{-i\xi n}e^{i\xi\frac{1}{2}}\Bigr]=\frac{1}{\xi i}\Bigl[-e^{-i\xi n}e^{-i\xi\frac{1}{2}}+e^{-i\xi n}e^{i\xi\frac{1}{2}}\Bigr]=\frac{2e^{-i\xi n}}{\xi}\Bigl[\frac{e^{i\xi \frac{1}{2}}-e^{-i\xi \frac{1}{2}}}{2i}\Bigr]=\frac{2e^{-i\xi n}}{\xi}\sin(\frac{\xi}{2})$

is this part correct?

b) prove that $\displaystyle (\hat{\phi_{n}})_{n=1}^{\infty}$ is an orthogonal system in $\displaystyle L^2(\mathbb{R})$. 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 $\displaystyle \langle \widehat{\phi}_n, \widehat{\phi}_m\rangle=0$ when $\displaystyle n\ne m$.

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

CB

Re: Fourier transform and orthogonal system

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

Parseval's theorem then tells you that $\displaystyle \hat{f}$ is orthogonal to all the functions $\displaystyle \hat{\phi}_n$.

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 $\displaystyle \hat{(\phi_{n})}$ with some other function and if

$\displaystyle \Bigl<\hat{(\phi_{n})},\hat{(\phi_{m})}\Bigr>$and control if this equals to 1 if m=n or 0 otherwise. Then we prove that $\displaystyle \hat{(\phi_{n})}$ is an orthogonal system right? but how do we choose /find the other function $\displaystyle \hat{(\phi_{m})}$??

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**rayman** 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 $\displaystyle \hat{(\phi_{n})}$ with some other function and if

$\displaystyle \Bigl<\hat{(\phi_{n})},\hat{(\phi_{m})}\Bigr>$and control if this equals to 1 if m=n or 0 otherwise. Then we prove that $\displaystyle \hat{(\phi_{n})}$ is an orthogonal system right? but how do we choose /find the other function $\displaystyle \hat{(\phi_{m})}$??

To find $\displaystyle \hat{\phi}_m$, you just replace n by m in the formula that you obtained for $\displaystyle \hat{\phi}_n.$ But to compute $\displaystyle \bigl\langle\hat{\phi}_n,\hat{\phi}_m\bigr\rangle$, you should use Parseval's theorem, which tells you that $\displaystyle \bigl\langle\hat{\phi}_n,\hat{\phi}_m\bigr\rangle = \bigl\langle\phi_n,\phi_m\bigr\rangle.$ That is much easier to calculate than the inner product of the Fourier transforms, and it will show you that the system $\displaystyle \hat{(\phi_{n})}$ is orthonormal. To show that it is not a complete system, you must find a nonzero $\displaystyle L^2$-function that is orthogonal to each $\displaystyle \hat{\phi}_n.$ The way to do that is again by using Parseval's theorem, as indicated in my previous comment.

Re: Fourier transform and orthogonal system

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Originally Posted by

**CaptainBlack** (a) looks OK

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

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

CB

It is easier to find a counter example than you might think, since almost any element of $\displaystyle L^2(\mathbb{R})$ that decays faster than $\displaystyle 1/\xi$ cannot be represented. (the space spanned by this set of functions is a subset of a set of periodic function of period $\displaystyle 2\pi$ times a $\displaystyle \rm {sinc}$ function - a bit of care and "set" can be replaced by "space" but I'm not minded to be careful today)

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.

Re: Fourier transform and orthogonal system

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Originally Posted by

**Hartlw** 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 $\displaystyle \exp(-{\rm{i}} n \xi), \ \ n= ... -1,0,1, ...$.

CB

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**CaptainBlack** 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 $\displaystyle \exp(-{\rm{i}} n \xi), \ \ n= ... -1,0,1, ...$.

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!

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.

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**Hartlw** 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

Re: Fourier transform and orthogonal system

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Originally Posted by

**Hartlw** 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 $\displaystyle \phi_n$ was incomplete. But that's ok, because if $\displaystyle \phi_n$ is incomplete, $\displaystyle \widehat{\phi}_n$ is incomplete because you can't create a complete basis using only an incomplete basis.

There is another way to show $\displaystyle \phi_n$ is incomplete using theorem from my previous post. Show (f,$\displaystyle \phi_n$) = 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 $\displaystyle \phi_n$. 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.

Re: Fourier transform and orthogonal system

Quote:

Quote:

Originally Posted by

**Hartlw** 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 $\displaystyle \phi_n$ are not all zeros for all $\displaystyle x$ less than some $\displaystyle x_0$, or rather you seem to be assuming that $\displaystyle n \in \mathbb{N}$ rather than $\displaystyle n \in \mathbb{Z}$ 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 $\displaystyle \{\widehat{\phi}_n, n\in \mathbb{Z}\}$, rather than $\displaystyle \{{\phi}_n, n\in \mathbb{Z}\}$. 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).

Quote:

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

There is another way to show $\displaystyle \phi_n$ is incomplete using theorem from my previous post. Show (f,$\displaystyle \phi_n$) = 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 $\displaystyle \phi_n$. 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 $\displaystyle \chi_{{[-1/4,1/4]}}$ cannot be represented in the basis $\displaystyle \{\phi_n, n\in \mathbb{Z}\}$, and so $\displaystyle \mathfrak{F} \chi_{[-1/4,1/4]}$ cannot be represented in the basis $\displaystyle \{\widehat{\phi}_n, n\in \mathbb{Z}\}$

CB

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**CaptainBlack** Quote:

Originally Posted by

**Hartlw** 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? (Worried)

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**Opalg** Has my comment #3 become invisible then? (Worried)

Your $\displaystyle f$ is $\displaystyle \phi_0$ under my assumption about the unspecified index set for the $\displaystyle \phi $s (but the basic idea can be made to work)

CB