# Help to understand wavelet basis on an interval proposed by Cohen, Daubechies and Vial.

I am reading Albert Cohen, Ingrid Daubechies, Pierre Vial. Wavelets on the Interval and Fast Wavelet Transforms, 1993.

In this paper wavelet bases on an interval (for, example on space $$L^2[0,1]$$) are proposed. They propose the same number of scaling functions and mother wavelets (namely, $$2^j$$ for $$j$$-th scale).

Firstly, a half-line $$[0,\infty)$$ is considered. Suppose we are dealing with wavelets which have minimal support $$[-N+1,N]$$ for number $$N$$ of vanishing moments (like Daubechies wavelets). Then the construction of scaling functions, as I understand it, is following:

1. For mother wavelets having $$N$$ vanishing moments, $$\int_{-\infty}^{\infty}a \psi(x)dx = 0$$ for $$a = const$$, so only scaling functions "generate" constants. This, however, is impossible on an (half-)interval (e.g. $$[0,\infty)$$), because a) $$\phi(x)$$ is continuous, b) $$\phi_{0,k}(0) = \phi_{0,k}(-k) = 0$$ for $$k \ge N-1$$ (here the first subscript index is scale index and the second is shift index). Thus, we add such a function $$\phi^{0}(x)$$ that it a) "generates" constants, b) is orthogonal to shifts of $$\phi(x)$$.
2. Define $$\phi^0(x)$$ as $$\phi^0(x) = 1 - \sum_{k=N-1}^{\infty} \phi(x-k)$$. This can be treated as a "constant" minus projections of that constant to all shifts of $$\phi(x)$$ which reside completely in $$[0, \infty)$$. Using $$\sum_{k=-\infty}^{\infty} \phi(x-k) = 1$$ we get $$\phi^{0}(x) = \sum_{k=-\infty}^{N-2} \phi(x-k) = \sum_{k=-N+1}^{N-2} \phi(x-k)$$. In the last equality I dropped all shifts of $$\phi(x)$$ which do not "live" on $$[0, \infty)$$ even partially.
3. Now we have an orthonormal set of scaling functions ($$\phi^0(x)$$ and $$\phi_{0,k}(x)$$ for $$k \ge N-1$$), but they still do not generate polynomials of the first degree (only constants). So we drop one (outermost) "internal" scaling function and add two new functions $$\phi^{0}(x)$$ and $$\phi^{1}(x)$$. I provide formulas for these functions from the article: $$\phi^{k}(x) = \sum_{n=k}^{2N-2} \binom{n}{k}\phi(x+n-N+1)$$ for $$k = 0, 1, \cdots, N-1$$

These functions, $$\phi^{k}(x)$$, $$0 \le k \le N-1$$ said to be independent, orthogonal to $$\phi_{0,k}$$, $$k \ge N$$ and have support $$supp \phi^{k}(x) = [0, 2N-1-k]$$.

I have the following questions:

• For N=2 (For example, Daubechies D4 wavelet), $$\phi^{0}(x) = \phi(x-1) + \phi(x) + \phi(x+1)$$ (using the formula above). Isn't its support $$[-2,3]$$, not $$[0, 3]$$? Support of $$\phi(x)$$ is $$[-1,2]$$, supp of $$\phi(x+1)$$ is, obviously, $$[-2, 1]$$ and supp of $$\phi(x-1)$$ is $$[0, 3]$$, resulting in $$[-2, 3]$$. Why the authors claim it is $$[0, 3]$$?
• Can you explain how $$\phi^{1}(x)$$, $$\phi^{2}(x)$$ and so on are constructed? $$\phi^{0}(x)$$ seems to be obvious to me, but not all others. As I understand $$\phi^{1}(x)$$ must be something like $$\phi^{1}(x) = x - \sum_{k=N}^{\infty} \left \phi(x-k)$$. How do we get the general formula for $$\phi^{k}(x)$$?