De Rerum Natura

I have started to work thru Anton Deitmar’s “A First Course in Harmonic Analysis”. This is background reading for my perusal of the Compressed Sensing literature. On the first chapter he wants to prove the convergence in the $L^2$-norm of Fourier series for periodic functions. To do this he asserts (offhandedly to my untrained eyes) the equality
$$\sum _{k=1}^n\cos (2\pi kx)=\frac{\sin ((2n+1)\pi x)}{2\sin (\pi x)}-\frac{1}{2}.$$

It is embarrassing how long it has taken me to prove this on my own. In the end, Euler’s formula,
$$e^{i\theta }=\cos \theta +i\sin \theta ,$$
ended up being the right thing to use since it turns the proof into checking the equality of polynomials. My proof is inductive. First prove the $n=1$ case,
$$\cos (2\pi x)=\frac{\sin (3\pi x)}{2\sin (\pi x)}-\frac{1}{2},$$
by rewriting it as,
$$\sin (\pi x)\cos (2\pi x)=\frac{1}{2}(\sin (3\pi x)-\sin (\pi x))$$
$$\left(\frac{e^{i\theta }-e^{-i\theta }}{2i}\right)\left(\frac{e^{i2\theta }+e^{-i2\theta }}{2}\right)=\frac{1}{2}(\frac{e^{i3\theta }-e^{-i3\theta }}{2i}-\frac{e^{i\theta }-e^{-i\theta }}{2i}),$$
the demonstration of the equality becomes a polynomial check. Induction to the case $n+1$ by using the same trick with Euler’s formula leads to the general proof of the formula.

Right off Wolfgang Pauli Strasse in ETH Hoeggenberg, the 8th Optical 3-D Conference is taking place from July 9-12. I’m here to present the paper on autonomous DEM uncertainty estimates. The conference is meant to highlight the increased convergence between photogrammetry and related sciences in computer vision, robotics, data-handling algorithms, etc. But the conference is mostly attended by photogrammetrers (?) with a sprinkling of other scientists.

My talk has been pigeonholed into a “Shape and Surface” technical session which is incongruous, but there really is not any other better choice. A lot of the sessions are on how to characterize and improve the precision of camera or laser scanner systems. Since my talk is about going the other way (many bad measurements = few good ones, figure out errors on the fly instead of the lab), I wonder how it will be received.