Hi All,

I wanted to make this post to address some of the concerns expressed in your emails regarding 'distributions' and 'functions'. This is covered extensively by online resources. I've had a brief look at the Wikipedia page and it seems fit for consumption. Please don't ask me anything about compact support. I don't know anything about it. Anyway, there are many different flavours of explanations and the preference of one over others differs between authors. I'm going to offer one of those flavours.

A distribution has some 'function-like' properties but not enough of those to make it a function. One of those properties is that it houses a class or family of functions AND, for each of those functions, points to a set of numbers (a field) that relates to that function. The class of functions that lie inside the distribution MUST be well-behaving. I think distributions are only accepted very reluctantly by mathematicians and recognition comes on the basis that (1) it is valuable as an analysis tool and (2) they have an explicit, analytic form.

Linearity is preferred but not necessary; see examples from stochastic calculus and operational calculus for additional explanation. This is not an issue for us because we only look at linear problems that do not involved coupled dependent variables and their derivatives. The only distributions we will EVER look at (maybe) are the PDF and the Dirac Delta. And yes, it is possible to multiply PDFs but there are constraints that limit those possibilities. I will talk more about these in class. There are a number of transforms that can change distributions to 'nicer' forms to allow for that operation.

I'll go through a few concrete examples during our next extension class on Tuesday 28 April. Email me if you have any special requests.