Is there a way to represent functions using circles, similar to how Taylor series work?

Question

I was curious whether or not there is a method of representing a function as an infinite amount of circles multiplied or added or something?

Similar to a Taylor series, such that the method would approximate the function.

Answer 1

Yes of course you can. Here is one example. Consider the template function

$$t\to f(t) = \sqrt{1-t^2},\,\,\, t\in[-1,1]$$

This is a half-circle. Now let us build a family of scaled and translated templates for $n\in \mathbb Z^+, k\in \mathbb Z$:

$$g_{n,k}(t) = f\left(\frac{t}{2^{n-1}}-k\right)$$

These will look something like this:

If we fit with a usual least-squares regression, we can approximate the Taylor basis functions on $[-1,1]$: