Articles of approximation theory

An apparently new method to compute the $n$th root of any complex number

I found  a series of articles (in Portuguese) by a Brazilian mathematician named Ludenir Santos, where presents a series of iterative methods, he said new, to extract nth roots of any complex number different from zero (and hence for real numbers), with any desired degree of accuracy.  In his first articles, he has two different methods […]

Runge's Theorem for meromrophic functions

Is there a name for this extension of Runge’s theorem? Theorem: Let $K\subset\mathbb{C}$ be compact, and let $A\subset K^c$ be a set which intersects each component of $K^c$. Let $f$ be meromorphic on an open set $U$ containing $K$. Let $A_f$ denote the set of poles of $f$ in $K$. Then there is a sequence […]

stone weierstrass approximation theorem for simple functions: does it exist?

The most general version of the theorem (I have seen) states that a function from a compact metric space into reals can be approximated by an algebra of functions that maps in the same way, is continuous, separates points, is non-vanishing. I wonder, apologies if the question is trivial (I am not a mathematician), what […]

Can Runge's approximating rat. fns. be required to take certain prescribed values?

Suppose $f$ is analytic on an open set $U$ containing the compact set $K$, and $\{r_n\}$ is a sequence of rational functions provided by Runge’s theorem (having poles in some prescribed set $A$). For a given finite set $\{x_1,\ldots,x_k\}\subset K$, can we additionally demand that for each $n$, $r_n(x_1)=f(x_1),r_n(x_2)=f(x_2),\ldots,r_n(x_k)=f(x_k)$? If yes, can we also require […]

Bounding error of Padé approximation

I’m trying to understand how one would understand the error of a given Padé approximation for a function. For instance, the $[2,1]$ approximant for $\log(1+x)$ is $\frac{x(6+x)}{6+4x}$. Is there a reasonable way to understand how good this approximation is for $|x|<1$ (without using the fact that we can actually go and calculate $\log(1+x)$ for any […]

Approximating piecewise linear function

I’m trying to derive an analytic approximation to the following piecewise linear function: $$ f(x) = \left\{ \begin{eqnarray} \frac{x}{x_s} & & \text{if} & x < x_s \\ \frac{1-x}{1-x_s} & & \text{if} & x \geq x_s \end{eqnarray} \right. $$ where (and this is an edit), $0\leq x\leq 1$ and $0<x_s<1$. I’ve been trying some approximating functions […]

References on Inverse Problems, Approximation theory and Algebraic geometry

For example, you approximate structure functions of finite simple graphs in cases where only cut sets of the systems are known. The inverse problem means to build possible scenarios in underdetermined system. Gröbner bases provides a way to express a set of structure functions — however how can you know that the set of structure […]

Every $K$-Lipschitz function can be uniformly approximated by $C^1$ functions with derivative bounded by $K$

The exercise states: Let $a,b\in\mathbb{R}$, $a<b$ and let $(C[a,b],\Vert\cdot\Vert)$ denote the vector space of continuous real functions on $[a,b]$ endowed with the uniform norm. Let $C^1[a,b]\subset C[a,b]$ denote the set of continuously differentiable functions and define $$ M_K=\{f\in C[a,b]: |f(x)-f(y)|\leq K|x-y|\quad\mathrm{for\;every\;}x,y\mathrm{\;in\;}[a,b]\}\\ A_K=\{f\in C^1[a,b]: |g'(x)|\leq K \quad\mathrm{for\;every\;}x\mathrm{\;in\;}[a,b]\} $$ Show that $\bar{A_K}=M_K$ I’ve shown that $M_K$ is […]

Proof of non-strictly convexity of $l_1$ and $l_{\infty}$

Given the Banach space $l_p = \left\{f\in R^n : \|f\| \leq \infty\right\}$ for $1\leq p \leq \infty$, we define the following norms $\|f\|_p = \left(\displaystyle\sum_{j=0}^{\infty} |f_j|^p\,w_j\right)^{\frac{1}{p}},\quad p\geq 1,\,w_j>0,\,\forall j\in\mathbb N$ $\|f\|_{\infty} = \underset{j\in\mathbb N}{\sup} |f_j|$ I want to prove that the spaces $l_1$ and $l_{\infty}$ are not strictly convex. For $l_1$, I’m trying to find […]

Interpolation, Extrapolation and Approximations rigorously

A foreign book mentioned that “when the Lagrange’s interpolation formula fails (for example with large sample due to Runge’s phenomenon), you should use approximation methods such as Least-squares-method.” I am confused because I have always thought that interpolation/extrapolations are approximations. My confusion lies in the fact that the book used the three terms as disjoint […]