Articles of approximation theory

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 […]

The magnitude of the difference between the integral and the Riemann sums for continuous functions

How do I prove that $$\lim_{n\to\infty}n\left[\int_{0}^{1}f(x){d}x – \frac{1}{n}\sum_ {i=0}^{n}f\left(\frac{i}{n}\right)\right] = -\frac{1}{2}$$ for any $ f(x)$ in $C^0([0,1])$ such that $f(0)=0$ and $f(1)=1$ ? Suggested steps: Set $f(x)=x+g(x)$; Prove that for any $g\in C^0([0,1])$ such that $g(0)=g(1)=0$, the above limit is zero$\ldots$ $\ldots$ by approximating $g$, uniformly over $[0,1]$, with a trigonometric polynomial of the form […]

Approximation of $L^1$ function with compactly supported smooth function with same mass and same uniform bounds

Recently, I have asked this question. Now, I even want to make this better. Given $f\in L^1(\mathbb{R})$ with $0\leq f\leq 1$, I can find for any $\epsilon>0$ a $g\in C_c^\infty(\mathbb{R})$ such that $\|f-g\|_{L^1}\leq \epsilon$. Can I assume wlog that $\|f\|_{L^1} = \|g\|_{L^1}$ and $0\leq g \leq 1$?

How does one derive Radial Basis Function (RBF) Networks as the smoothest interpolation of points?

I was reading/watching CalTech’s ML course and it said that one could derive the RBF Gaussian kernel from the solution to smoothest interpolation that minimizes squared loss. i.e. one can derive the predictor/interpolator: $$ f(x) = \sum^{K}_{k=1} c_k \exp( -\beta_k \| x – w_k \|^2 )$$ from the Empirical Risk minimizer with a smoothest Regularizer: […]

maybe this sum have approximation $\sum_{k=0}^{n}\binom{n}{k}^3\approx\frac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty$

prove or disprove this $$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$ this problem is from when Find this limit $$\lim_{n\to\infty}\dfrac{\displaystyle\sum_{k=0}^{n}\binom{n}{k}^3}{\displaystyle\sum_{k=0}^{n+1}\binom{n+1}{k}^3}=\dfrac{1}{8}?$$ first,follow I can’t $$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$ .Thank you for you help

What is “Approximation Theory”?

What exactly is “Approximation Theory”? If I read the wikipedia-article I doesn’t get much clearer. Why are “pure” mathematicians interested in it? I see a lot of people that do harmonic analysis also do approximation theory.

Is Ramanujan's approximation for the factorial optimal, or can it be tweaked? (answer below)

Ramanujan’s famous factorial approximation, $$n!\approx\sqrt{\pi}\left(\frac{n}{e}\right)^n\root\LARGE{6}\of{8n^3+4n^2+n+\frac{1}{30}}$$ is far more accurate that the Stirling approximation when applied to numbers less than 1, but it’s not quite there. What could be done to tweak it without adding to its complexity? Update: After experimenting with Ramanujan’s approximation, I’ve discovered this disturbing fact: apparently, the error decreases slower than the […]

How to best approximate higher-degree polynomial in space of lower-degree polynomials?

My question is: Find the best 1-degree approximating polynomial of $f(x)=2x^3+x^2+2x-1$ on $[-1,1]$ in the uniform norm(NOT in the least square sense please)? Orginially, as the title of the post suggests, I’m asking the general problem: given an $n$-th degree polynomial $p(x)$ on $[-1,1]$, find the best approximation $q(x)$ in lower degree($j<n$) polynomial space: $\min_{\operatorname{deg}q(x)=j} […]