# Questions tagged [approximation-theory]

Approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby.

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### What is a good way to sample data points on an interval as to avoid Runge's phenomenon but have a deterministic sampling scheme?

I recently asked: Is there an analogous Gibbs phenomena to approximating sinusoidal but with polynomial terms? because I noticed that at the edges, polynomial interpolation of equidistance points ...
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### Pade approximation of a rational function

I believe I have a naive or hard question because I couldn't find any results in the Internet yet. Any help is greatly appreciated. So suppose I have two rational functions $R_1(x)$ and $R_2(x)$, i....
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### Extend the Stone-Weierstrass theorem to high dimension?

I am thinking of if there is high dimensional extension to the well known Stone-Weirstrass theorem. Wikipedia says it is possible to extend the 1D theorem to 2D, i.e. If  f  is a continuous real-...
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### How to approximate a negative exponential distribution?

In Hierarchical Web Caching Systems: Modeling, Design and Experimental Results by Hao Che, Ye Tung and Zhijun Wang, the authors used $$g(t) = \frac{\exp(- (t - \tau)/(T - \tau))}{T - \tau}$$ to ...
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### What is the exact meaning of 'in the first approximation' in the context of applied mathematics?

In the applied mathematics textbooks or papers, I often see the phrase 'in the first approximation'. For example, substitution of Eq.(1) into the boundary condition (2) results in Eq.(3) describing ......
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### Is there an analytic approximation of the following function

Let $\psi$ be a function on $\mathbb{R}$ satisfying $\phi(x)\geq 0$ for any $x$ and $\psi(x)=0$ when $|x|\geq 1$. $\int_{-1}^1\psi(x)dx=1$ $\int_{-1}^1x\psi(x)=0$. $|\psi'''(x)|\leq B$ for a ...
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### Numerical Approximation Rules

I was not exactly sure what to title this question, but I would appreciate if someone can confirm my understanding of the left hand, right hand, trapezoidal, midpoint, and simpsons approximations. ...
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### Control higher derivatives of approximating sequence

Let $\Omega \subset \mathbb R^d$ be a bounded open set. Let $H_0^1(\Omega)$ be the usual Sobolev space. From the definition it follows, that each $u \in H_0^1(\Omega)$ can be approximated by a ...
Suppose we want to approximate a function $f(x)$ on a interval $[c,d]$ by, say, a linear polynomial $p(x) = a_0 + a_1x$ using the scalar product $$\langle f,g\rangle = \int_a^bw(x)f(x)g(x)\,dx,$$ ...