Feeds:
Posts

## Mittag-Leffler function of two parameters related to Besov norm/fractional Hilbert transform?

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

10:24 PM (0 minutes ago)

 to David, bcc: aimee
Dear Professor Donoho,

In order to better understand what’s going on with these Mittag-Leffler 2-parameter families, I look at two objects: Hilbert transform which is convolution with 1/pi x and also the Besov norm in the snapshot standard definition.  I am very rusty on analysis so I need time to understand this situation.  I studied once a course with Elias Stein and I read Singular Integrals and Differentiability properties of functions years ago but this situation requires a better grasp for me about what these waiting times could be doing.  I can simply blindly fit the Mittag-Leffler functions to data as well, but it is difficult to know what of nature is being captured here without understanding the relation to something I can understand from analysis.  Besov norms have a power parameter in the denominator.  We know that on the statistical denoising side if the signal is Besov then wavelet shrinkage recovers it.  This Mittag-Leffler 2-parameter family, assuming it would fit a distribution of waiting times empirically, would be telling us via Ito formula with a distribution that is not quite a power law that the noise level to threshold is some function of the second term of Ito’s formula applied to f(x)=x^2.  Therefore there is some relation between Besov norm denoising results and these Mittag-Leffler deviations from pure power law.  These circle of ideas are grasping at some finer tuned understanding of smoothness in the small of natural phenomena which is something wholly deep and quantitative and has little to do with the smoothness that humans think of.  Frankly, these things happen in my mythology because of the constraints of a compact spherical four-spatial dimensional universe — compactness being key but that is far beyond my reach even by intuitive leaps.  Besov norm, empirical Ito density and the variations of Mittag-Leffler functions all seem to be different paths to some feature of finer quantitative study of volatility (clearly because here we can compute the Ito density easily).
2 Attachments

Preview attachment Screenshot from 2015-06-10 22:03:33.png

Screenshot from 2015-06-10 22:03:33.png

Preview attachment Screenshot from 2015-06-10 22:23:27.png

Screenshot from 2015-06-10 22:23:27.png