If $a_1$ to $a_3$ is the solution of this linear system of equations $$\left(\begin{array}{rrr|r} -1&2&-3& -1\\ 1&-8&27&0 \\ -1 & 32 & -243 & 0 \end{array}\right)$$

then $f_3(x) = a_1\sin(x)+ a_2 \sin(2x) + a_3 \sin(3x)$ defines a function with $f_3(\pi) = 0$ and the convergence rate of Newton's methods should locally be of 6th order since $f_3'(\pi) = -1$ and the 2nd to 6th derivatives vanish at $\pi$ (That's how the system of equations is constructed). As far as I know from my class on numerical methods, this is sufficient for the rate of convergence.

Similarly, one can extend the defining system of equations above to construct functions which imply an arbitrary rate of convergence for Newton's method by computing coefficients $a_1$ to $a_k$ with $f_k(\pi) = 0$, $f_k'(\pi) = -1$ and $f_k^{(i)}(\pi) = 0$ for $2 \leq i \leq 2k$. The convergence rate of Newton's method is therfore of the order $2k$.

Numerical calculations in Matlab confirm these results.

By plotting the functions $f_k$ one can observe that the function seems to converge pointwise to $g(x) = \pi -x $ for $x \in (0, 2\pi)$.

**If this is true, how can one prove this?**

I'm not familiar with functions definied by solutions of linear systems of equations and have never seen something like this before.

I tried to use the Taylor expansion of $|f_k(x)| = |\frac{1}{(2k+1)!} \cdot f^{(2k+1)}(\xi)| \gtrapprox a_k\frac{k^{2k+1}}{(2k+1)!} (x-\pi)^{2k+1}$.

But $\frac{k^{2k+1}}{(2k+1)!}$ diverges when $k$ approches $\infty$, so I either need an estimation of $a_k$ or another approach to show pointwise convergence.

Is there any material about functions defined by linear system of equations or closely related topics (preferably undergraduate level)? Did anyone, by chance, have a smiliar idea to this and/or is there any further material available on this topic?

**Is it connected to the Fourier series?**

Another question I asked myself when I discovered these functions was if it is connected to the Fourier series of $\pi-x$ (extended periodically beyond $[0, 2\pi]$) because it is a linear combination of sine functions with integer factors in the argument. I remember that if there exits a uniformly convergent series of the form $\sum_{k= -\infty}^\infty c_k e^{ikx}$ it must be the Fourier series. We can write $f_k = \sum_{i= 0}^k a_{ki}\sin(ix)$, so there can't be any coefficients $a_{\infty i}$ such that the sum defined by $f_\infty$ converges uniformly to $\pi-x$ because that would imply the uniform convergence of the Fourier Series of $\pi-x$, which it does not, if I remember correctly.

Can one define something like $a_{\infty i}$ in a reasonable way? And if so, are they the Fourier coefficients?I have no idea how to start here. I have found some results about infinite systems of linear equations but nothing directly applicable to my problem.

Thanks in advance for any answers.