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You do not specify your class of $f$, and most importantly, do not specify where the convergence should hold, for which $x_0$. In general, the answer on your question is Newton's method: $g(x)=x-f(x) …

You can look at my linear algebra lecture notes http://www.math.purdue.edu/~eremenko/dvi/fft2.pdf and .../fft.pdf

I know the references for question 4: MR3659421 Schleicher, Dierk; Stoll, Robin Newton's method in practice: Finding all roots of polynomials of degree one million efficiently. Theoret. Comput. Sci. 6 …

If $\phi(x)=x^m$ your function is analytic at $0$, and convergence rate can be found with very high precision. Same happens if it is analytic in sufficiently large sector with vertex at $0$. See about …

Let $f(x)={\mathrm{erf}}(x)-\tanh(x)$. It can be easily seen from Taylor series at $0$ and from asymptotics at $\infty$ that $f(x)>0$ for small $x$ and for large $x$. Let us prove that $f(x)>0$ by co …

With your understanding of "positive", the answer is negative. Take a closed bounded totally disconnected set $E\subset R$ of positive measure. Let $f$ be the characteristic function of this set. I cl …

This question has been studied in two papers of Peter Yuditskii and myself: Zbl 1241.41005 (arXiv:1008.3765) and Zbl 1168.30020 (arXiv:math/0604324), where we determined the polynomial of best approxi …

Edited after the comment of Joel David Hamkins. The "correct $n$-th digit of a real number, of of real and imaginary part of a complex number is ill defined. For example, what is the $n$-th digit of t …

The theory of approximation in the complex plane is almost as rich as the theory on the real interval. Some of the good books are: D. Gaier, Lectures on complex approximation. Translated from the Germ …

Orthogonalize your powers on your interval using the Gram-Schmidt algorithm, and then apply the usual Fourier formulas for expansions in an orthogonal system.

The answers to the first series of questions are negative since a continuous function in general CANNOT be uniformly approximated by polynomials of $z$ uniformly on a set with non-empty interior: inde …

This problem has an exact solution, written in the book N. I. Akhiezer, Theory of approximation. Dover Publications, Inc., New York, 1992, Chap II section 37. The error is $$\frac{(1-\sqrt{1-\rho^2})^ …

Chaplygin's theorem is true even for non-linear differential equations of the form $$y^{(n)}=F(x,y,y',\ldots,y^{(n-1)}),$$ under the only assumption that both $y$ and $z$ their derivatives $z$ exist …

There is a characterization of Schwarzian derivatives of rational maps: section 3 in the text: http://www.math.purdue.edu/~eremenko/dvi/schwarz3.pdf There is something similar also in arXiv:math/0512 …