I am reading an article on wavelet connection coefficients (G. Beylkin, "[On the representation of operators in bases of compactly supported wavelets](https://epubs.siam.org/doi/10.1137/0729097)", 1992 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=1191143))) and I came across Equation (3.31):
\begin{equation}
\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}
\end{equation}
where $\phi(x)$ is the scaling function and
\begin{equation}
M\_l^\phi = \int\_{-\infty}^\infty x^l\phi(x)\,dx
\end{equation}
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
\begin{equation}
\sum\_{l=-\infty}^\infty lr\_l = -1
\end{equation}
where 
\begin{equation}
r\_l = \int\_{-\infty}^\infty \phi(x-l)\phi'(x)\,dx.
\end{equation}




---


EDIT: Using the Poisson summation as @Nemo suggested in a [comment](https://mathoverflow.net/questions/354714/wavelet-momentum-identity#comment889964_354714), I was able to find that
\begin{equation}
\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}.
\end{equation}


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?



Wavelet momentum identity
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,](http://cas.ensmp.fr/~chaplais/FTP/Mathematical_Notes/Strang_and_Fix.pdf) 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.](https://books.google.nl/books?id=idIIAQAAQBAJ&pg=PA258) (The identity is equivalent to the recursion relation 4.51.)


