As my loyal reader knows, Kate Storey-Fisher (NYU) and I are looking at the Landy–Szalay estimator for the correlation function along a number of different axes. Along one axis, we are extending it to estimate a correlation function that is a smooth function of parameters rather than just in hard-edged bins. Along another, we are asking why the correlation function is so slow to compute when the power spectrum is so fast (and they are equivalent!). And along another, we are consulting with Alex Barnett (Flatiron) on the subject of whether we can estimate a correlation function without having a random catalog (which is, typically, 100 times larger than the data, and thus dominates all compute time).
Of course when you get a mathematician involved, strange things often happen. One thing is that Barnett has figured out that the Landy–Szalay estimator appears in literally no other literature other than cosmology! And of course even in cosmology it is only justified in the limit of near-zero, Gaussian fluctuations. That isn't the limit of most correlation-function work being done these days. In the math literature they have different estimators. It's clear we need to build a testbed to check the estimators in realistic situations.
One thing that came up in our discussion with Barnett is that it looks like we don't ever need to make a random catalog! The role that the random catalog plays in the estimation could be played (for many possible estimators) by an auto-correlation of the survey window with itself, which in turn is a trivial function of the Fourier transform of the window function. So instead of delivering, with a survey, a random catalog, we could perhaps just be delivering the Fourier transform of the window function out to some wave number k. That's a strange point!
In the discussion, I thought we might actually have an analytic expression for the Fourier transform of the window function, but I was wrong: It turns out that there aren't analytic expressions for the Fourier transforms of many functions, and in particular the Fourier transform of the characteristic function of a triangle (the function that is unity inside the triangle and zero outside) doesn't have a known form. I was surprised by that.
Although there isn't an analytic expression for the Fourier transform of a triangular window function there is a simple analytic form for the FT of the gradient of the window function (which is just three Dirac delta line segments). For example see:
ReplyDeletehttps://math.stackexchange.com/questions/2431048/fourier-transform-of-a-triangle
Essentially the FT of a triangle gradient is the sum of three corrugated sinc functions with different orientations.
The problem with non analytic forms frequently arises in areas like reconstruction from Radon projections and reconstruction from differential phase images.
In my experience it is then often possible to keep all the analysis really simple by just considering the gradients of functions you are interested in.