Approximating/describing smooth bounded function, that has infinite derivatives, on [-1..1]

I have a practical problem.

I have continuous functions defined on [-1..1] with [-1..1] bound. The functions might have infinite derivatives like $\sqrt x$ has at 0.

One example:

I want to approximate those functions.

Sorry for non-mathematical descriptions, but here are the desirable properties of the approximation system:

• The approximation function should be easily evaluated at given x.
• More parameters - better approximation (in $L_2$ or $L_1$ norm sense)
• Diminishing returns - each parameter improves the approximation less than the previous one did
• The domain of each parameter is some real interval (maybe infinite) and each combination of parameters define a valid approximation function.
• It would be best if the range of parameters is finite though.
• The approximation function should not be [much] more complex than the original function (say, in terms of the number of critical points). I fear that truncated Taylor or Fourier series would exhibit such problem.

I could have used something like a 2D Bezier curve for approximation, but not all parameter combinations define a valid function.

P.S. I'm not actually interested in the approximation per se, but rather in an opposite problem: Defining a continuous parameter space that contains points representing approximations of elementary functions and their combinations.

• Can you provide more details about the function that you would like to approximate? All we know is that the function is continuous and besides that can behave very badly... Commented Oct 1, 2017 at 2:49
• I want explore the parameter space of the approximations of "elementary" functions and their combinations, squeezed to a $[-1..1]x[-1..1]$ boundary box using sigmoid-like transforms. I won't say that these functions behave badly. I might be able to approximate the above function using only 3 2D Bezier points. Actually, I'm not interested in the approximations per se, but rather in an opposite problem: Define a continuous parameter space that contains points representing approximations of elementary functions and their combinations. Commented Oct 1, 2017 at 3:03

The first candidate for the "space" you're referring to would be $P_n$ -- the space of polynomials of degree at most $n$ on $[-1,1]$. When constructing approximations, the best basis of this space is typically the set of Chebyshev polynomials. You get better and better approximations by including Chebyshev polynomials of higher and higher degree. One problem, though -- polynomials don't work very well when the approximant has a vertical slope. So, you may have to use rational functions, instead.