# 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.

717 questions
Filter by
Sorted by
Tagged with
25 views

### Approximate formula from data (7 inputs, 1 output)

I'd like to approximate a formula which 'fits' my data. The formula should take 7 inputs and produce a single ouput. The input variables are listed in columns A to G of my data (see image below). The ...
84 views

### Approximation of dynamic systems

I was looking for a formal approach to simplify models of dynamic systems. Say we have a dynamic system given by $\frac{dx}{dt} = f(t,x,u), ~~~~x(t_0) = x_0$ $y = g(x)$ We know $f$ and $g$ but ...
118 views

### Zolotarev number and commuting matrices

Recently in a post (link) upper bounds on the singular values $\sigma_j(X)$ of a matrix $X$ have been considered. To restate the central observation, it says that if $AX−XB=F$ for $A$ and $B$ normal ...
71 views

### Show that there exists 2 different best approximations

Let $F$ be a linear space with a norm $\lVert \cdot \rVert$ which is not strictly convex. Show that there is a function $f \in F$ and a subspace $S \subset F$, so that $f$ has different best ...
2k views

### Proving that: $9.9998\lt \frac{\pi^9}{e^8}\lt 10$?

I would like to use mathematical tools to prove that $$9.9998\lt \frac{\pi^9}{e^8}\lt 10$$ With an on-line calculator I got $$\frac{\pi^9}{e^8}\approx 9.9998387978$$ But I do not know any ...
119 views

47 views

88 views

33 views

### Does the function $U=\frac{kx}{(x+x_{0})^2}$ reduce to a simple harmonic potential energy function for small oscillations around $x=x_0$?

The question is from a physics problems book, A particle of mass $m$ moves in a potential energy function given by $$U=\frac{kx}{(x+x_{0})^2}$$ where $x$ denotes the position and $x_0$ is a ...
3k views

### An error formula for linearization

Question: Can anyone shed some light on this formula? I can't find any information on it. It has three corollaries that I also need to understand:
169 views

### What is the relation between Chebyshev and Taylor polynomials?

I just read about Chebyshev polynomials and that they are used in approximations. I don't fully understand them yet. What is the relation between Chebyshev polynomials and Taylor expansions?
Suppose we have $K$ points $x_1,\ldots,x_K$ in $\mathbb{R}^d$ and let $$f(x)=\sum_{k=1}^K \exp(-\lambda \Vert x-x_k \rVert^2).$$ Can we uniformly bound $f$ independent of $K$? It is okay to use ...