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I am currently at Princeton for the conference “The power of Analysis” honouring Charlie Fefferman‘s 60th birthday. I myself gave a talk at this conference entitled “Recent progress on the Kakeya conjecture”; I plan to post a version of this talk on this blog shortly.
But one nice thing about attending these sorts of conferences is that one can also learn some neat mathematical facts, and I wanted to show two such small gems here; neither is particularly deep, but I found both of them cute. The first one, which I learned from my former student Soonsik Kwon, is a unified way to view the mean, median, and mode of a probability distribution on the real line. If one assumes that this is a continuous distribution for some smooth, rapidly decreasing function with , then the mean is the value of that minimises the second moment
the median is the value of that minimises the first moment
and the mode is the value of that maximises the “pseudo-negative first moment”
(Note that the Dirac delta function has the same scaling as , hence my terminology “pseudo-negative first moment”.)
for the fractional differentiation operators applied to a sufficiently nice real-valued function (e.g. Schwartz class will do), in any dimension and for any ; this should be compared with the product rule .
The proof is as follows. By a limiting argument we may assume that . In this case, there is a formula
for some explicit constant (this can be seen by computations similar to those in my recent lecture notes on distributions, or by analytically continuing such computations; see also Stein’s “Singular integrals and differentiability properties of functions”). Using this formula, one soon sees that
and the claim follows.