Constructing a function - Fourier Series?

[FreeGamer]

Homework Statement

Construct a function that is infinitely differentiable, f(x) in [0,1] for all x, and f(x)=1 for -1<x<1, f(x)=0 for |x|>2.

Homework Equations

None.

The Attempt at a Solution

I thought of doing it using a Fourier series for a square wave, in the way that f(x)=1 for -1.5<x<1.5, but since the function is not periodic, I would have to somehow make it so that f(x)=0 for |x|>2.

Now what I'm not sure is if this function

f(x)= { Fourier series of the square wave from -1.5 to 1.5 } ( for |x|<=2 ), 0 ( for |x|>2 )

would still be infinitely differentiable in such setting, in particular at the point x=2 and x=-2. If this is not the way to do it, can someone please hint on a different path.

Thanks in advance!

 

[vela]

What do you mean by

Construct a function that is infinitely differentiable, f(x) in [0,1] for all x

specifically "in [0,1] for all x"?

 

[Dick]

No, it's nothing to do with Fourier series. The trick is to connect a function that's constant on an interval with another function that's not while still making it infinitely differentiable. Define f(x)=exp(-1/x) for x>0 and f(x)=0 if x<=0. Can you show that's infinitely differentiable? Have you done something like this in class? If not look at "Smooth transition function" in http://en.wikipedia.org/wiki/Non-analytic_smooth_function It's an example of the sort of thing you are looking for.

 

[FreeGamer]

What do you mean by

specifically "in [0,1] for all x"?

By that I mean the values of f(x) must be between 0 and 1 for all x, sorry if i confused u a bit.

No, it's nothing to do with Fourier series. The trick is to connect a function that's constant on an interval with another function that's not while still making it infinitely differentiable. Define f(x)=exp(-1/x) for x>0 and f(x)=0 if x<=0. Can you show that's infinitely differentiable? Have you done something like this in class? If not look at "Smooth transition function" in http://en.wikipedia.org/wiki/Non-analytic_smooth_function It's an example of the sort of thing you are looking for.

Thanks! I think Prof mentioned something about it but never seriously discussed about it. I think I should be able to prove the differentiability either myself or with help on the wiki site.