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I only wanted to say, on the one hand, bravo on the sine page in your private nlab web; on the other hand, the main inspiration for the goofy inequality was that I wanted a similar presentation to that of the natural exponential function which, as everyone knows who knows it, satisfies
and furthermore that this pins it down exactly.
Cheers!
I see; thanks for the explanation! And sorry for the rudeness (“goofy”); I’ll get rid of it.
So this characterization of the sine is due to you? Very interesting; I’d never seen that before. You must have devised your own proof; I’d be interested in hearing it!
I don’t mind “goofy”.
Goodness, I just lost a lot of editing, and I’ve got the last assignment marking of the year still to do…
The Tricky Part of the argument is to consider that the function we want is of the form $g(x) = x h(x)$, and then construct an equivalent functional equation for $h$:
$h(x) = h(x/3) - \frac{4}{27} x^2 h(x/3)^3$iff $g (x) = 3 g(x/3) - 4 g(x/3)^3$.
In terms of the (nonlinear nonlocal) transformation $T$
$T : f \mapsto x \mapsto f(x/3) - \frac{4}{27} x^2 f(x/3)^3$one calculates
$(T F - T f)(x) = (F - f)(x/3) \left( 1 - \frac{4}{27} x^2 (F F + f F + f f) (x/3) \right)$which shows
Now consider the particular bounds $F_0 = 1$ and $f_0 (x) = 1-x^2$. An otherwise uninteresting calculation gives
$T F_0 (x) = 1 - \frac{4}{27} x^2$ $T f_0 (x) = 1 - \frac{ 16 }{ 27 } x^2 + x^4 P(x)$for an explicit polynomial $P$; in brief,
$f_0 \le T f_0 \le T F_0 \le F_0$on some interval $[-r,r]$, which need not be bigger than $[-1,1]$. This is the start of an induction argument that
$f_0 \le T^n f_0 \le T^{n+1} f_0 \le T^{n+1} F_0 \le T^n F_0 \le F_0$while at the same time (induction via item 2) we have the bounds
$| T^n F_0 (x) - T^n f_0 (x) | \leq \frac{x^2}{9^n} .$It follows that $T$ has a unique fixedpoint within the specified bounds, over the interval $[-r,r]$, and hence a unique fixedpoint over the whole real line. So that’s existence and uniqueness. Since, obviously, the functional equation and the bounds are concocted to hold for sine, we might be happy with that.
Thanks very much, Jesse – I think I get the overall idea; I can run over this with a fine-toothed comb maybe a little later. (That 4/27 looks weirdly suggestive…)
But where on earth does all this come from? Did you find this characterization in a book somewhere, or what? It looks just a bit off the beaten track, shall we say, at least to my eyes.
I wanted to impress on some calculus students just how much easier everything is with the right tools; so, here is a complete characterization of a familiar-ish thing, but what on earth can you do with it? But after developing some calculus, e.g. Taylor series, one can start crunching digits of things like $sin(1)$, or prove that it’s irrational, and so forth.
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