Sequence f_(n+1) = sin^2(pi/2*f_n) on [0,1]

[mollwollfumble]

Have you seen this sequence before?
Does this sequence have a name?

I stumbled across this piece of mathematics by accident this morning.
It's something I've been looking for for about thirty years.

Consider the sequence:
f_0 = x
f_1 = sin^2(pi/2*x)
f_2 = sin^2(pi/2*sin^2(pi/2*x))
f_3 = sin^2(pi/2*sin^2(pi/2*sin^2(pi/2*x)))
...
f_(n+1) = sin^2(pi/2*f_n)

These functions f_i all have the properties:
1) Continuously differentiable to all orders (i.e. the ultimate in
"smooth")
2) f_i(0) = 0
3) f_i(1) = 1
4) f_i(0.5) = 0.5
5) As i tends to infinity, f_i tends to a step function
6) Monotonic
7) f_i tends to a step function very quickly.

See graph on
http://i120.photobucket.com/albums/o162/DavidPaterson/SSSF/Sine_Splines.gif

I really wish I'd known about this ages ago. They can be used as
blending functions. Terms in this sequence could be used in creating
transition curves for making sense of experimental data. They could be
used for splines. There are applications in signal processing.
Possibly even in Finite Elements and probability distributions.

[
Moderators note:
This uses the computer-language convention for notation,
pi/2*x = (pi/2)*x
and not the mathematicians' convention
pi/2*x = pi/(2*x)
]

[dro...@gmail.com]

On Dec 15, 4:31�am, mollwollfumble <David.Pater...@csiro.au> wrote:
> Have you seen this sequence before?
> Does this sequence have a name?
>
> I stumbled across this piece of mathematics by accident this morning.
> It's something I've been looking for for about thirty years.
>
> Consider the sequence:
> f_0 = x
> f_1 = sin^2(pi/2*x)
> f_2 = sin^2(pi/2*sin^2(pi/2*x))
> f_3 = sin^2(pi/2*sin^2(pi/2*sin^2(pi/2*x)))
> ...
> f_(n+1) = sin^2(pi/2*f_n)
>
> These functions f_i all have the properties:
> 1) Continuously differentiable to all orders (i.e. the ultimate in
> "smooth")
> 2) f_i(0) = 0
> 3) f_i(1) = 1
> 4) f_i(0.5) = 0.5
> 5) As i tends to infinity, f_i tends to a step function
> 6) Monotonic
> 7) f_i tends to a step function very quickly.
>

> See graph onhttp://i120.photobucket.com/albums/o162/DavidPaterson/SSSF/Sine_Splin...

>
> I really wish I'd known about this ages ago. They can be used as
> blending functions. Terms in this sequence could be used in creating
> transition curves for making sense of experimental data. They could be
> used for splines. There are applications in signal processing.
> Possibly even in Finite Elements and probability distributions.
>

There is an old problem in Polya and Szego (Problem 173, volume 1,
Chapter 4, Vermischte Aufgaben, in German edition)
which says that if sin(x) > 0, then (sqrt(n/3))sin(sin(sin . . .(x)..)
-> 1. (Function sin iterated n times)
This result implies a lot of your observations, since you essentially
take iterations of the function sin(sin((pi/2)*x).
By examinig the solution of the problem, you can probably deduce
everything you ask for.

As ever,

Vladimir Drobot

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
* Vladimir Drobot
* Retired and gainfully unemployed
* http://www.vdrobot.com
* mailto:dro...@pacbell.net
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-

[mjc]

I think this is also discussed in de Bruijn's "Asymptotic Methods in
Analysis".