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# 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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### “Taylor series” is to “Volterra series” as “Padé approximant” is to _________?

Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it. Volterra ...
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### $\operatorname{frac}(na)$ is dense in $[0,1]$ for irrational $a$. How to find the smallest $n$ efficiently?

It is well known that the sequence $\operatorname{frac}(na)$ , where $\operatorname{frac}$ denotes the fractional part and $n$ runs over the positive integers, is dense in $[0,1]$. Suppose, I have ...
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### What is the rational function that deviates least from $0$?

It is a well known result that among the set of polynomials of degree $n$ with leading coefficient $1$, the $n^{th}$ order Chebyshev polynomial $T_n (x)$ minimizes the (point-wise maximum) deviation ...
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### An accessible reference to proof of Nachbin's theorem

Stone-Weierstrass theorem. Let $X$ be a compact Hausdorff space, let $\mathscr{A}$ be the algebra of real valued continuous functions on $X$ and let the topology in $\mathscr{A}$ of uniform ...
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### Approximation theory in Lp spaces (Reference Needed)

I am looking for some reference on approximation theory in Lp spaces. I have found a number of papers like: paper1 , paper2 etc. I was wondering if there is a book or a monograph that will contain ...
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### Generalizations of equi-oscillation criterion

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...
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### Approximating a Discrete Function with a Polynomial

Consider a discrete function $f : \{x_1,\dots,x_n\} \to [a,b]$ that, for some $\omega>0$, satisfies $$\max_i |f_i-f_{i-1}| \le \omega. \tag1$$ Let $q$ be the polynomial of order $m<n$ that, ...
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### $||\phi-\phi_\epsilon||_{L^1(\mu)}<\epsilon$ and $||\phi-\tilde{\phi}_\epsilon ||_{L^1(v\mu)}<\epsilon$, then $\tilde{\phi}_\epsilon= \phi_\epsilon$?

Let $\Omega\subset \mathbb{R}^n$ be an open and bounded set. Let $\mu:\mathcal{B}(\Omega)\to [0,+\infty)$ a bounded Radon measure and let $\varphi, \, v \in L^1(\Omega,\,\mu)$, $v\geq 0$. Then $v\mu$ ...
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### How far apart can L1 and L2 lines fit to the same data be?

Given $n$ points $(x_i, y_i)$ in the unit square with $x_i = {{i - 1} \over {n - 1}}$ uniformly spaced and $0 \leq y_i \leq 1$, consider the best-fit L1 line and the best-fit L2 line: $\qquad$ L1 ...
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### Convergence rate of multivariate polynomials.

Given a continuous function $f:[a,b]\rightarrow\mathbb{R}$, it is a well known result that, for each $n\geq0$, there exists a polynomial $p_n$ that best approximates $f$ in $\|\cdot\|_\infty$ among ...
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### 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 ...
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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 ...
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### The approximation function of $\frac{x}{y}$

Is there a approximation function of $$\frac{x}{y},$$ and the approximation function is in the form of $f(x) + f(y)$ or $f(x) - f(y)$. That's to say the approximation function can split $x$ and $y$.
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### Does the minimax polynomial exist in the multivariate case?

The following theorem can be found in many lectures on approximation theory: Theorem: Let $f$ be a continuous function defined on the interval $[a,b]$. Then there exists a polynomial $p_n\in\Pi_n$ ...
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### Approximation of “smooth” discrete functions

Assume $f : \{1,\ldots,n\} \to \mathbb{C}$ satisfies $|D^\ell f(i)| \leq C$ for all $i \in \{1,\ldots,n-\ell\}$ and all $\ell \in \{0, \ldots, k\}$ for some $k \in \mathbb{N}$. Here, $D$ denotes ...
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### Calculate integral curve through a discretely sampled vector field

given a finite sample of a smooth vector field, i.e. a set $$S=\{(p_i,v_i)\mid i\in \{1,\dots, n\}, p_i,v_i\in\mathbb{R}^2\}$$ where the $p_i$ are the points at which we have a vector $v_i$, how do I ...
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### Time Discretization

I wonder why we work with constant discretization in Time Discretization of numerical approximation for numerical scheme if we take not necessarily constant Discretization Is the numerical scheme (...
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### approximating an integral/hypergeometric function

I am looking to approximate the following integral for small $z$: $\int_0^{\infty}dy \frac{1}{z} e^{-y/z} \frac{w e^{-y}}{s + w e^{-y}}$ . The integral can be solved in general to be a ...
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### Chebyshev Equioscillation Theorem in $L_{\infty}[a,b]$?

Let $a,b\in\mathbb{R}$, $a<b$. Consider \begin{align} C[a,b] & :=\{f\in\mathbb{R}^{[a,b]}:f\text{ is continuous}\}\text{,} \\ L_{\infty}[a,b] & :=\{f\in\mathbb{R}^{[a,b]}:f\text{ is ...
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### Error term when Lagrange interpolating continuous non-differentiable functions

Suppose I know the values of a continuous function on $[a,b]$ in some finite number of points $x_0,x_1 \ldots x_n$. I can form the Lagrange interpolating polynomial, $p$. I am curious if there is any ...
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### Interpolation error of $C^k$ function derivatives

Consider the interval $I=[-1,1]$ and the set of points $A=\{r_1, ..., r_n\}\subset I$. Suppose that there exists a $k$-times continuously differentiable $f\in C^k(I, \mathbb{R})$ of which we know its ...
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### Generalizing the Runge approximation theorem for Riemann Surfaces: approximating by nonvanishing functions

The Runge approximation theorem for open Riemann Surfaces says the following. If $X$ is an open Riemann surface and $Y \subset X$ is a Runge subset, i.e., $X \setminus Y$ has no relatively compact ...
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### Problem on othogonal polynomials with Lebesgue mesure

This is an over my level assignmet i got: Let $\mu$ denote the Lebesgue (of arc-length) measure on the unit circle $\mathbb{T}:=\{|z|=1\}$. Prove that for the measere $dv(z)=|1-z|^{2}d\mu$ the ...
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### Can $x^n$ be uniformly approximated by the combination of $x^{k^2}$?

For each $n\in\mathbb{N}$ , can $x^n$ be uniformly approximated by the linear combination of $\left(x^{k^2}\right)_{k\in\mathbb{N}}$ ? In order to facilitate a solution, we might as well try to ...
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### Does the Sobolev space $W^{1,p}(\Omega), p>2$ has a monotone basis?

A Shauder basis in a Banach space is monotone if $\|P_{n}f\|\leq\|f\|,$ where $P_{n}$ is the projection to the sum of the first n elements of the basis. For Hilbert spaces this is always the case if ...
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### Do two close functions share some local minima?

Let $f,g:\mathbb{R}^n \to \mathbb{R}$ be two differentiable functions. Assume that $\| f -g \|_{\infty} \leq \epsilon$. On what conditions on $f$, for every local minima $x$ of $g$, there is a local ...
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### tan nx is not as popular as Chebyshev polyomials?

I noticed that $\tan(nx)$ can be written as the ratio $P(\tan x) / Q(\tan x)$, where $P$ and $Q$ are polynomials. Indeed, I found a short proof here Expansion of $\tan(nx)$ in powers of $\tan(x)$ ...
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### Polynomial approximation in Sobolev spaces

Let $U \subseteq \mathbb{R}^n$ be an open connected set. Let $p>1$, and let $f \in W^{1,p}(U)$. I have recently heard that $f$ can be locally approximated in $W^{1,p}$ by polynomials. That is, ...
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### approximate bijective function such that the inverses are bijective and “easily” computable

I have a infinitely differentiable, bijective function $f:[0,1]\to[0,1]$, and I would like to approximate this function by a series of other functions $T_i$ (think Taylor) – with the conditions that ...
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### Curve fitting N points with n(fixed) quadratic curves

I essentially have a constrained curve fitting problem that I need to solve efficiently. The following problem arises when performing practical calibration of RSSI (signal strength), providing ...
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### Why does the sup norm make the results of approximation theory independent from the unknown distribution of the input data?

I was reading the paper "Why and When Can Deep – but Not Shallow – Networks Avoid the Curse of Dimensionality: a Review" and I was trying to understand the following statement in section 3.1: On ...
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$H_p^s$ is a standard $L^2$ Sobolev space of periodic functions on $[0, 2\pi]$, with $\|f\|^2_{H_p^s} = \sum_{j = 0}^s \|f^{(j)}\|_{L^2}^2$. Let $$f_N(x) = \sum_{|n| < N} \hat{f}_n \frac{1}{\... 0answers 35 views ### Bounding sum of samples of a Gaussian Suppose we have K points x_1,\ldots,x_K in \mathbb{R}^d and let$$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 ...
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....
I'd like to know more about some patterns I found in graphs corresponding to irrational numbers. Here's the graph for $\sqrt 2$ for example First, I'll try to explain most naturally the function that ...