Results 1 to 2 of 2

Math Help - Haar Wavelets

  1. #1
    Junior Member
    Joined
    Nov 2009
    From
    Pocatello, ID
    Posts
    59

    Haar Wavelets

    Show that the rescaled Haar wavelets \psi_{jk}(x)=2^{j/2}\psi(2^jx-k) form an orthonormal basis basis for L^2(\mathbb{R}).

    So I know that what I have to show is that:

    \int_{\mathbb{R}}\psi_{jk}(x)\psi_{lm}(x)dx=\delta  _{jl}\delta_{km}

    I'm just stuck as to how to simplify that integral.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Hmm. Maybe this might help:

    \psi_{jk}(x)=2^{j/2}\begin{cases}<br />
1,&\quad 0\le 2^{j}x-k<1/2\\<br />
-1,&\quad 1/2\le 2^{j}x-k<1\\<br />
0,&\quad\text{otherwise}<br />
\end{cases}=2^{j/2}\begin{cases}<br />
1,&\quad k\le 2^{j}x<1/2+k\\<br />
-1,&\quad 1/2+k\le 2^{j}x<1+k\\<br />
0,&\quad\text{otherwise}<br />
\end{cases}

    =2^{j/2}\begin{cases}<br />
1,&\quad 2^{-j}k\le x<2^{-j}(1/2+k)\\<br />
-1,&\quad 2^{-j}(1/2+k)\le x<2^{-j}(1+k)\\<br />
0,&\quad\text{otherwise}<br />
\end{cases}.

    Here I'm using the mother wavelet function \psi(x) as defined in the wiki. So the only points at which this function is nonzero are in the half-open interval [2^{-j}k,2^{-j}(k+1)).

    Now, what if you could show that [2^{-j}k,2^{-j}(k+1))\cap[2^{-l}m,2^{-l}(m+1))=\varnothing if either j\not= l or k\not=m? That would certainly be sufficient to show the zero part of the delta functions, wouldn't it? Because then, under the integral sign, each function would drag the other one down to zero.

    What happens when both j=l and k=m? What does the integrand do?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Pulse response and Phase function (image processing-Wavelets)?
    Posted in the Advanced Math Topics Forum
    Replies: 3
    Last Post: December 6th 2010, 02:14 AM
  2. Stability of an impulse response (image processing -Wavelets)?
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: December 3rd 2010, 05:26 PM
  3. foorier series from haar function..
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 16th 2010, 11:13 PM
  4. The Mathematical Theory of Wavelets
    Posted in the Math Topics Forum
    Replies: 0
    Last Post: December 12th 2009, 03:45 AM
  5. wavelets
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: March 29th 2009, 12:44 PM

Search Tags


/mathhelpforum @mathhelpforum