(08/09/2014, 07:02 PM)jaydfox Wrote: I'm also working on a second-order approximation. This would consist of two parts. The first is to fix oscillations in the error term (1/1.083...)*f(k)-A(k). These oscillations follow approximately a scaled version of f(-x). Remember the zeroes on the negative real line?

First of all, in thinking about it, the oscillations actually seem to follow something along the lines of G(f(-c x^2)), where G(x) is some scaling function (similar to x^p for some 0 < p < 1, or perhaps similar to asinh?), and c is a constant. It's not exact of course, but I think it gets me in the right ballpark. I'll show pictures later to demonstrate the similarities that I'm working with.

The -x^2 indicates a couple things. The negative sign means I want to look at the oscillations that are clearly visible in f(x) on the negative real axis. However, between each zero is an alternating local minimum or local maximum. The oscillations I'm looking at show both a local minimum and a local maximum at the same scale.

I.e., if x_k is a zero (e.g.,-1054.13), then approximately 2*x_k is another zero (e.g.-2359.66). In between, we have either a local minimum or a local maximum. However, with the oscillations of A(k)-f(k), we would have a local minimum and a local maximum over the same doubling of scale, and thus two "zeroes". We're working with a discrete sequence, so there won't actually be any zeroes. Anyway, the x^2 term takes care of this. It means that in going from x to 2x (technically, (2+h)x for a small h), we're looking at zeroes between -x^2 and -4x^2 (technically, -(4+4h)x^2 for the same h, ignoring the O(x) term), and thus we've considered two zeroes. (I'm having trouble putting it into words at the moment, so maybe some pictures later on will help.)

Okay, here are some pictures to illustrate the oscillations I was talking about. There are couple things to note. First of all, the self-similarity repeats at slightly larger than powers of 2. For example, the local maxima for the even terms (top "curve") are at 4, 10, 28, 68, 156, 348, 780, etc., each more than twice the previous index, but less than 3 times. The ratio gets close and closer to 2, though it take quite a while to get down as low as 2.1, let alone 2.01.

Second, while there is definitely some self-similarity, the vertical scale is not linear. While I am slightly more than doubling the scale in the x direction with each picture below, the y scale increases by much larger factors, up to 1000x per image. That's where the unknown G(x) above comes into play.

~ Jay Daniel Fox