# Wavelet momentum identity

I am reading an article on wavelet connection coefficients (G. Beylkin, "On the representation of operators in bases of compactly supported wavelets", 1992 (MSN)) and I came across Equation (3.31): $$$$\sum_{l=-\infty}^\infty l^m\phi(x-l) = x^m + \sum_{l=1}^m (-1)^l \begin{pmatrix} m\\l \end{pmatrix} M_l^\phi x^{m-l}$$$$ where $$\phi(x)$$ is the scaling function and $$$$M_l^\phi = \int_{-\infty}^\infty x^l\phi(x)\,dx$$$$ is the $$l$$-th momentum of $$\phi$$.

The author claims that the equation is well-known if $$M_l^\phi = 0$$ for $$l=0,\dotsc,m$$, and the general case follows from taking Fourier transforms. However, I could not find it, and trying to prove it myself is not working.

I recognize that both sides are kinds of convolutions, but when taking the Fourier transform the expressions (apparently) lead nowhere. Is there some trick I need to be aware of, or is it simply lack of practice/knowledge?

I do not know if this information should help, but the ultimate goal is to prove that $$$$\sum_{l=-\infty}^\infty lr_l = -1$$$$ where $$$$r_l = \int_{-\infty}^\infty \phi(x-l)\phi'(x)\,dx.$$$$

EDIT: Using the Poisson summation as @Nemo suggested in a comment, I was able to find that $$$$\sum_{l=-\infty}^\infty l^m\phi(x-l) = \sum_{k=0}^m \sum_{l=-\infty}^\infty (-1)^k \begin{pmatrix} m\\k \end{pmatrix} e^{-ilx} i^k \frac{d^k\hat{\phi}}{d\xi^k}(-l) x^{m-k}.$$$$

Now, I know that $$i^k\frac{d^k\hat{\phi}}{d\xi^k}(0) = M_l^\phi$$ but I'm still stuck with the terms $$i^k e^{-ilx} \frac{d^k\hat{\phi}}{d\xi^k}(-l)$$ for $$l\ne0$$. Is there any identity I am not aware of?

• mathworld.wolfram.com/PoissonSumFormula.html – user82588 Mar 11 at 21:44
• I used it by defining $f(y) = y^m\phi(x-y)$ and almost found the result. However, instead of $M_\phi^l$, I got $i^k e^{i l x} \frac{d^k\hat{\phi}}{d\xi^k} (-l)$, which (I think) I cannot turn into $M_\phi^l$... – AspiringMathematician Mar 13 at 13:28

The identity in the OP does not hold for any $$m$$, but only for $$m< N$$ where $$N$$ is the number of vanishing moments of the wavelet function. To complete the Poisson-summation derivation, one needs the socalled Strang-Fix condition, which says that $$\frac{d^k\hat{\phi}}{d\xi^k}(-l)=0$$ for integer $$l$$ unequal to 0 and $$k< N$$.
The identity says that integer shifts of the scaling function can reproduce polynomials of order $$N$$. For a proof, see theorems 4.26 and 4.27 in this book. (The identity is equivalent to the recursion relation 4.51.)