# Equality between two equations

at the moment I am reading the following paper

Benno, Steven A., and José MF Moura. "On translation invariant subspaces and critically sampled wavelet transforms." Multidimensional Systems and Signal Processing 8.1-2 (1997): 89-110.

The step between the equations (10) and (11) I can't comprehend. The step is as follows

$$\int_R G(\omega) \overline{\tilde{G}(\omega)} e^{-j2\pi\omega\tau} \Big(\sum_k e^{-j2\pi(f-\omega)k}\Big)d\omega =$$ $$\int_R G(\omega) \overline{\tilde{G}(\omega)} e^{-j2\pi\omega\tau} \Big(\sum_k \delta(f-\omega+k)\Big) d\omega.$$

I know that $$e^{-i2\pi k}$$ for $$k\in\mathbb{Z}$$ is an orthnormal basis, but not over $$\mathbb{R}$$ and I have no idea, why $$k$$ comes into the dirac function with an "+". At most I woud expect something like $$\delta(f-\omega)$$ since it is in a product with k.

Clearly there is a typo in equation (10), the correct expression is $$\int_R G(\omega) \overline{\tilde{G}(\omega)} e^{-j2\pi\omega\tau} \Big(\sum_k e^{-j2\pi(f-\omega+k)}\Big)d\omega,$$ which can be seen by substituting the definition of $$a_k$$ exactly as the authors describe.
• Hi, thank you for your answer. I checked equation (10) if $a_k$ is inserted. The sum is reordered by k and (10) should be correct. – Matthias Lauber Jan 6 '19 at 11:19