Carrying on with topic of a mollifier family. Actually, I want to talk about blending functions. They are, for me, the more interesting aspect of mollifiers.

A blending function is a function like g(x)=exp(-1/x) for x positive, and g(x)=0 for x zero or negative. The aspect of a blending function that is useful in defining a mollifier family is that the blending function is smooth, ie has derivatives of all orders, at its “blend” point of x=0. By defining a mollifier function f_A(z)=g(A-abs(z)) or similar structure, times a suitable normalizing coefficient, one obtains a “blip” function around the point Z which is nonzero only on a radius-A ball around the point z. This blip function f_A can be used as a smooth test function to average (by convolution) the values of an observation function (provided by data or by theoretical model) around z. Because f_A is nonzero only within a ball of radius A (it has bounded support, is the terminology) one can make various manipulations that are most convenient with integrals over finite regions. Applications include electrodynamics (Maxwell mentions blending functions), and statistical mechanics or particle physics — see the writings of C.N. Yang and T.D. Lee, or Wigner and Eisenbud — where it is convenient to make a hypothesis, for the development of the model, that one is working with functions which are bounded in their effect. To some extent the assumption of a blending function is a given. However, I’ve become interested in the blending functions themselves. So I’m going to focus on families of blending functions, not applications such as the definition of mollifier families.

I had an unfortunate mishap with WordPress’s post editor. I had written quite a bit of info, and making reference to taking of limits, wrote the character sequence dash-dash-greaterthan. What I had forgotten is that sequence of three characters is used to denote a comment in HTML coding. When I saved my post as a draft, the editor discarded the comments! Or rather, what it thought were comments. Very disappointing. Anyway, I’m going to put my thoughts on blending functions into a Latex/PDF document and post it to a website. Will link from here in another post.

I will mention here very briefly the core idea. Given a polynomial Q(y) which has degree at least 1, and positive top coefficient (ie, coefficient of highest nonzero term). Consider the set H of functions P(y)/exp(Q(y)) where P(y) is any polynomial. That set is closed under differentiation (try it out for yourself), and of course it is closed under multiplication by another polynomial in y. All those functions tend to zero as y tends to infinity, because the polynomial in the numerator is dominated by the exponential in the denominator. The prototype is h(y)=1/exp(y), which is the set H defined by Q(y)=y, and the particular numerator P(y)=1, but there are of course a whole family of functions h() in the set H.

Given a nonzero member h() of the set H, one can define the function g(x)=h(1/x) for positive x, and g(x)=0 for x zero or negative. Following the same line of reasoning as in the John Loftin writeup (linked via prior post), one proves that the functions g(x) are smooth, notably at x=0. So they are blending functions. Some care is needed to be sure that g(x) based upon arbitrary nonzero h() cannot be always zero after some number of differentiations (but that is true), and to verify that g(x) is eventually of one sign in a small enough neighborhood of x=0 (but that also is true), and to word the definition of blending function carefully enough to make the reasoning flow well (but that can be done). It is longer than I wish to risk trying to post via the WordPress editor! But you can work it out yourself in the meantime.

Then, one might wish to consider other functions in the denominator. For example, what about the set H of functions of the form P(y)/(exp(y)+const) which is getting into something like Fermi-Dirac integral behavior, seen for instance in the free electron gas model of metals.

Enough for now. I’m going to see if I can get the website set up. Will let you know a URL when have it ready.

Best wishes,
Ken Roberts
19-May-2014