ChapterZero

This morning, I was working on what is probably my favorite math problem of those posted to the UHME mailing list. I haven’t taken a crack at it in eons, and have never made a crack in it, so I forgot exactly what the problem was, and ended up making a calculation that I thought might be useful. Turns out it’s not; I have no use of the fact that

.

Despite that, it’s nice to know one more pretty identity that I’ll never use . What may be marginally more useful is a decimal expansion, but I haven’t yet been able to find a way to get one. That fractional part makes this such a tricky problem!

I think the answer is 0, just because it seems to me likely that this problem has to be solved by contradiction.

We’ve reached the midpoint of the semester in the Fourier class: the mid term is next week. I have free choice in taking it or not, by the simple expedient of showing up or not. I haven’t decided if I want to; on the one hand, I’d have a more meaningful assessment of my progress than the homework, but on the other, I’d have to study :). I’ve gotten accustomed to not doing any studying…

Here’s an interesting concept: approximate identities. Those have been used ingeniously in class to avoid explicitly proving many convergences, ones that we spent so much effort on in my Real Analysis class. The concept we’re using is a little different (less abstract, I suppose, but otherwise conformal to the standard definition, I believe) since they are naturally defined in general Banach algebras, but we’re only dealing with the particular Banach algebra . Here’s our definition:

A family of elements in is called an approximate identity for if

1. 
2. 
3. for every

The usefulness of an approximate identity is that it is guaranteed— by basic calculus— that as , so as a result, you have pointwise convergence of the sequence to almost everywhere.

So far, that has trivialized the proof of the uniqueness of the Fourier transform in , and the inversion theorem, (since the Fejer kernel— not quite the same as the one linked to, but the gist is the same— is an approximate identity) which I’m all for. They also seem suspiciously like the delta function ‘formalisms’ that EE students are introduced to: a series of functions with compact support that shrinks down while the area under the functions remain constant. I suppose this is to be expected, since the ‘real’ identity in the algebra would be the delta function, since it satisfies for all functions … except the delta function is not really a function. So we approximate it instead.