Let $\delta, \varepsilon \in (0,1)$. I am interested in a sequence $\{ \tilde{f}_N\}_N$ of polynomial approximations of the square root function $x \to x^{1/2}$ on $[\delta,1]$, of the form $$ \tilde{f}_{N}(x) = \sum_{i=0}^N \alpha_i x^i $$ which satisfies \begin{align*} \sup_{x \in [\delta,1]} \left|x^{1/2} - \tilde{f}_N(x)\right| & \leq \varepsilon, \\ N & = O\left(\frac{1}{\delta} \log\frac{1}{\varepsilon}\right),\\ \sum_{i=0}^N |\alpha_i| & \leq B, \\ \end{align*} where $B$ is a universal constant independent of $N$. (Obviously, I also want the sequence to satisfy $\tilde{f}_{N+1}(x) = \tilde{f}_{N}(x) + \alpha_{N+1} x^{N+1}$.)

Does such a sequence of approximations exist?

We can relax the requirement that B is a constant, and it would also be interesting if it could be taken as poly($N$).

More generally, I am interested in such a sequence of approximations, which satisfy the same properties, for the function $x \to x^{\alpha}$ where $\alpha \in (0,1)$.

Let $\delta, \varepsilon \in (0,1)$. I am interested in a sequence $\{f_n\}$ of polynomial approximations of the square root function $x \to x^{1/2}$ on $[\delta,1]$, of the form $$ f_n(x) = \sum_{i=0}^n \alpha_i x^i $$ which satisfies \begin{align*} \forall x \in [\delta,1],\ \left|f_n(x) - x^{1/2}\right|\ & \leq \varepsilon \\ n & = O\left(\frac{1}{\delta} \log\frac{1}{\varepsilon}\right),\\ \sum_{i=0}^\infty |\alpha_i| & \leq B, \\ \end{align*} where $B$ is a universal constant. (Obviously, I also want the sequence to satisfy $f_n(x) = f_{n-1}(x) + \alpha_n x^n$.)

Does such a sequence of approximations exist?

We can also relax the requirement that $B$ is a constant, and require $\sum_{i=0}^n |\alpha_i|$ to have a bound which is polynomial in $n$.

More generally, I am interested in approximations satisfying the same properties for the function $x \to x^{\alpha}$ where $\alpha \in (0,1)$.

Let $\delta, \varepsilon \in (0,1)$. I am interested in a sequence $\{ \tilde{f}_N\}_N$ of polynomial approximations of the square root function $x \to x^{1/2}$ on $[\delta,1]$, of the form $$ \tilde{f}_{N}(x) = \sum_{i=0}^N \alpha_i x^i $$ which satisfies \begin{align*} \sup_{x \in [\delta,1]} \left|x^{1/2} - \tilde{f}_N(x)\right| & \leq \varepsilon, \\ N & = O\left(\frac{1}{\delta} \log\frac{1}{\varepsilon}\right),\\ \sum_{i=0}^N |\alpha_i| & \leq B, \\ \end{align*} where $B$ is a universal constant independent of $N$. (Obviously, I also want the sequence to satisfy $\tilde{f}_{N+1}(x) = \tilde{f}_{N}(x) + \alpha_{N+1} x^{N+1}$.)

Does such a sequence of approximations exist?

More generally, I am interested in such a sequence of approximations, which satisfy the same properties, for the function $x \to x^{\alpha}$ where $\alpha \in (0,1)$.

Let $\delta, \varepsilon \in (0,1)$. I am interested in a sequence $\{ \tilde{f}_N\}_N$ of polynomial approximations of the square root function $x \to x^{1/2}$ on $[\delta,1]$, of the form $$ \tilde{f}_{N}(x) = \sum_{i=0}^N \alpha_i x^i $$ which satisfies \begin{align*} \sup_{x \in [\delta,1]} \left|x^{1/2} - \tilde{f}_N(x)\right| & \leq \varepsilon, \\ N & = O\left(\frac{1}{\delta} \log\frac{1}{\varepsilon}\right),\\ \sum_{i=0}^N |\alpha_i| & \leq B, \\ \end{align*} where $B$ is a universal constant independent of $N$. (Obviously, I also want the sequence to satisfy $\tilde{f}_{N+1}(x) = \tilde{f}_{N}(x) + \alpha_{N+1} x^{N+1}$.)

Does such a sequence of approximations exist?

We can relax the requirement that B is a constant, and it would also be interesting if it could be taken as poly($N$).

More generally, I am interested in such a sequence of approximations, which satisfy the same properties, for the function $x \to x^{\alpha}$ where $\alpha \in (0,1)$.