Suppose $f(x)$ is a real-valued function of a real variable that is continuous, smooth and monotone increasing at least on $x\in[0,\infty)$, satisfying the boundary condition $f(0)=0$. We will pick $f(x)=4\unicode{x200a}x^2+x$ as an example (there is nothing particularly important about this specific function, it is just simple and fits our purposes).
Suppose we want to find a function $g:\mathbb R^+\to\mathbb R^+$ that is continuous, smooth and monotone increasing on $x\in[0,\infty)$, and satisfies $g\circ g=f$, or, stated in a more verbose form, $g(g(x))=f(x)$. Such a function is called a functional square root$^{[1]}$ (also known as compositional square root or half-iterate) of $f$, and is sometimes denoted as $g=f^{\small[1/2]}$. In general, a functional square root of a given function is not unique, although sometimes there are natural additional conditions (such as analyticity) that can make it unique.
We want to find a functional square root of $f(x)=4\unicode{x200a}x^2+x$. One obvious approach is to look for a solution in the form of a formal power series $f^{\small[1/2]}(x)=\sum_{n=0}^\infty c_k\,x^k$, and then find the coefficients $c_k$ by calculating the composition of this series with itself and balancing corresponding coefficients on both sides. Indeed, this approach yields an infinite series $$f^{\small[1/2]}(x)=x+2\unicode{x200a}x^2-4\unicode{x200a}x^3+16\unicode{x200a}x^4-80\unicode{x200a}x^5+432\unicode{x200a}x^6-2304\unicode{x200a}x^7+\dots,$$ where all coefficients are integers that can be easily computed$^{[2]}$. Alas, the absolute values of the coefficients grow quite fast, and the radius of convergence of this power series is zero, meaning that we cannot use this formula to compute actual values of $f^{\small[1/2]}(x)$. We can only hope that it will give an asymptotic series for $f^{\small[1/2]}(x)$ near zero, if we are able to construct that function eventually.
So, we need another approach. Perhaps, we can construct $f^{\small[1/2]}(x)$ initially on some interval (say, $[0,1]$) as the limit of its increasingly precise polynomial approximations. Let's define $\rho_n(x)$ as a polynomial of degree at most $n$, monotone increasing on $[0,1]$, that minimizes $\delta_n=\max\limits_{\small x\in[0,1]}\left|\rho_n(\rho_n(x))-f(x)\right|$. Our idea is that by increasing $n$ we provide more degrees of freedom to the polynomial, so we expect that its composition with itself will be able to fit $f(x)$ more closely, so, hopefully, $\lim\limits_{n\to\infty}\delta_n = 0$.
Now we can define $f^{\small[1/2]}(x)=\lim\limits_{n\to\infty}\rho_n(x)$ at those $x\in[0,1]$ where the limit exists.
Numerical experiments suggest that this method indeed seems to work, and $\delta_n$ steadily and quickly decreases when $n$ goes up. This leaves us with several questions that at this point I do not know how to approach, so I am asking for your help.
- Is $f^{\small[1/2]}(x)=\lim\limits_{n\to\infty}\rho_n(x)$ well-defined on the whole interval $[0,1]$?
- Is it continuous?
- Is it smooth (class $C^\infty$)?
- Is it real-analytic at any point of the interval?
- If we perform a similar construction of $f^{\small[1/2]}$ using a larger interval (say, $[0,2]$), will the result agree with the one constructed on $[0,1]$?
- Is the Taylor expansion of $f^{\small[1/2]}(x)$ near zero gives the same divergent formal power series as we found above? Can any methods for regularization/resummation of a divergent series be used to construct $f^{\small[1/2]}(x)$ from its formal power series expansion?
