Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I think this is asked as a standard exercise in books about wavelets (e.g. exercise 7.2 in Mallat's book), but I couldn't find a proof. Let $\phi$ be a scaling function (see definition below). I would like to learn why $$\sum_{k\in\mathbb Z} \phi(x-k) = 1 $$
almost everywhere.

Definition. A sequence of subspaces $\{V_j: j\in \mathbb{Z}\}$ of $L^2(\mathbb R)$ is called a multiresolution analysis if it satisfies the following:

  • $V_j \subset V_{j+1}$
  • $\bigcap_{j}V_j = \{0\}$
  • $\overline{\bigcup_jV_j} = L^2(\mathbb R)$
  • $f(x)\in V_j$ if and only if $f(2x) \in V_{j+1}$
  • There exists a function $\phi \in V_0$ such that $\{\phi(x-k)\}_{k\in\mathbb Z}$ is an orthogonal basis for $V_0$

The function $\phi$ here is called as a scaling function.

share|improve this question

1 Answer 1

up vote 2 down vote accepted

I think you will find the proof for this in Mallat 1989, 'Multiresolution approximations and wavelet orthonormal bases of L^2'. Theorem 1 (in particular Equations (23), (36)) is what you are after. It is not trivial, longer to prove than I immediately thought. Perhaps there is a very fast proof but I can't think of it now.

share|improve this answer
    
@Glen: But $1\notin L^2$. If it were, a formal argument would be $1 = \sum c_k \phi(x-k)$, and $c_k = \int 1\cdot \phi(x-k) = $const. Some tricks are required to make it rigorous. This is not homework, you may give an outline if you have an idea. –  AgCl Dec 7 '10 at 16:17
    
@AgCl: What was in my mind when I wrote that was possibly using a sequence of scaled characteristic functions on longer and longer intervals, something like f_l = 2^-l\chi_{[-l,l]}. But now I think about it some more, this doesn't seem to work. I think I should just delete my answer until I think of a better idea, yes? –  Glen Wheeler Dec 8 '10 at 10:01
    
@AgCl: OK. I just looked it up in a paper, Mallat 1989, 'Multiresolution approximations and wavelet orthonormal bases of L^2'. Theorem 1 (in particular Equations (23), (36)) is, I think, what you are after. It is far from immediate, longer to prove than I immediately thought. Perhaps there is a very fast proof but I can't think of it now. I could reproduce the proof there if you wish, although it would probably be better to just read the paper. –  Glen Wheeler Dec 8 '10 at 10:56
    
@Glen: Thanks for the reference, that seems to be what I was looking for! Could you edit your answer with a reference to this paper, so that I can accept it? –  AgCl Dec 8 '10 at 23:31
1  
For lazy people like me... –  J. M. Dec 9 '10 at 10:04

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.