Power series and Fourier identity approximated in two or three iterations

Question

I understand that Fourier has proven that the sum of sines and cosines can be used to describe (almost) any curve. The power series describe that the sum of polynoms can be used to describe (almost) any curve.

I'd like to provide a simple to understand example to non-math students, so I am looking for a curve f(x) and their representation in polynoms p(x) (what students already understand) as well as the representation in sines and cosines t(x).

The function f(x) needn't be known. It is ok if it exists in painted form only. But of course it would be easier if it is known, simply for plotting it in Maxima. f(x) should be less than 10% off from both representations within the view port.

The representations p(x) and t(x) should have three parts maximum so that the students can actually follow. I don't want to go into summation of unlimited series.

You may choose the view port. Of course it would be easier to follow if the viewport is from -5 to 5 or something similar like in usual school homework.

Since I have some problems to express myself here in English, see this expected result:

If

• p(x) is as simple as $ ax^3 + bx^2 +cx + d$ and
• t(x) is as simple as $a*sin(bx+c) + d*sin(ex+f)$ and
• both are not too far off from f(x)

then I hope I have made a good step making them trust me, that this can be made more accurate by adding more terms.

The point I want to make to the students it:

1. you can record any signal, resulting in f(x)
2. it is possible to approximate using polynoms (p(x)) <- they know polynoms, so they can follow
3. it is also possible to approximate using sines and cosines (t(x)) <- although they don't know, they can still imagine that it is possible

Alright, so anyone did understand what I need and could give me a good example for p(x) and t(x) or at least a clue on where to start solving that puzzle?

For plotting purposes, f(x) should be $\frac{p(x)+t(x)}{2}$, right?

Answer 1

Take a well-behaved function like e.g.

$$f(t)=(t^2-1)^2\arctan[3(t+0.5)],\quad t\in[-1,1]$$

which can be approximated quite well with only a few series coefficients. A reasonably good power series approximation of this function is

$$p(t)=0.94732+0.89870t-2.65643t^2-1.02788t^3+1.80793t^4$$

A Fourier series approximation with a comparable approximation error is

$$q(t)=0.42182-0.50747\cos[\pi(t+1)]-0.31072\sin[\pi(t+1)]$$

This is how the function and its approximations look like: