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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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11
votes
0answers
381 views

A generalization of an integral related with $\zeta(2)$

It is well-known that: $$ \int_{0}^{+\infty}\frac{x}{e^{x}-1}\,dx = \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} \tag{1}$$ but what is known about $$ I_2 = \int_{0}^{+\infty}\frac{x^2}{e^x-1-x}\,dx \...
10
votes
0answers
122 views

Is there a theory of “almost symmetry” generalizing group theory?

Apologies for the inescapably soft question. Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost ...
10
votes
0answers
379 views

About the Gibbs phenomenon for the Legendre polynomials as an orthogonal base of $L^2(-1,1)$

In a recent question, I proved that the Fourier-Legendre expansion of the function $f(x)=\text{sign}(x)$ over $(-1,1)$ is given by: $$2\sum_{m\geq 0}\frac{4m+3}{4m+4}\cdot\frac{(-1)^m}{4^m}\binom{2m}{...
7
votes
0answers
90 views

“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 ...
7
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0answers
116 views

$\operatorname{frac}(na)$ 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 ...
6
votes
0answers
68 views

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 ...
5
votes
0answers
177 views

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 ...
5
votes
1answer
120 views

Polynomials with specified ranges in intervals

Say I have two finite intervals $[a,b],[c,d]\subsetneq\Bbb R$ where $a<b<c-1<c<d$ and $b-a=d-c=s<1$. I want to find a polynomial $f \in \Bbb R[x]$ such that $$\forall x\in[a,b],\mbox{ }...
4
votes
2answers
87 views

Monotonicity of function involving ratios of modified Bessel function of first kind

I am a scientist who is trying to come up with some analytic solutions for a system that I only have approximate answers to and I have run into the problem of proving that the following function is ...
4
votes
0answers
88 views

Convergence speed of discrete approximation

Here I asked the question about approximating the function $g(x) := \mathbb{E}(f(x,Y))$, where $x \in R$ and $Y$ is a random variable. If you follow the link you will see that $g(x)$ can be ...
4
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0answers
117 views

Converse of Taylor's Theorem

Let $n$ be a nonnegative integer and $a,b\in\mathbb{R}$ such that $a<b$. From Taylor's Theorem, we know that any $n$-time differentiable function $f:(a,b)\to \mathbb{R}$ satisfies the condition ...
4
votes
0answers
141 views

Using the Saddle point method (or Laplace method) for a multiple integral over a large number of variables

I am trying to understand the saddle point method used in the large N limit of matrix models. First, for the case of the integral of a single variable I found this notes There they say that you can ...
4
votes
0answers
155 views

Weighted polynomial approximation on the half-line

Let's $w : \mathbb R_+ \to \mathbb R_+^*$ a continuous function (I will only be interested in the case $w : t \mapsto e^{-t}$). We use $w$ to define the Banach space $C^0_w(\mathbb R_+)$ of continuous ...
4
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0answers
63 views

who discovered (or can be cited for writing) $\cos^n x$ is very close to $\exp\left(\frac{-nx^2}{2}\right)$ over $[-\frac{\pi}{2}, \frac{\pi}{2}]$?

In the course of working on some correlation problem, I happened to notice that in the vicinity of $0$, $\cos^n x$ is a pretty darned good approximation to $\exp\left(\frac{-nx^2}{2}\right)$. In ...
4
votes
0answers
92 views

polynomial approximation for $\frac{1}{1-x}$

Prove that for every $n\in\mathbb{N}$ there exists a polynomial $P$ such that $\deg P=n$ and $$ \sup_{x\in\left[0,\frac{1}{2}\right]}\left| P(x)-\frac{1}{1-x}\right|\ <\ \frac{4}{(\sqrt{2}+1)^{2n+...
4
votes
0answers
154 views

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 ...
4
votes
0answers
56 views

Is there a way to approximate a polynomial as another, binary-coefficient polynomial?

Let's say I have a polynomial: $$p(x) = \sum_{n=0}^N a_n x^n$$ where $x \in \mathbb C$. Does there exist theory and/or methods on approximating $p$ as: $$p(x) \approx \hat p(x) = \sum_{m=0}^M b_n x^m$...
4
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0answers
307 views

Approximation of Sets of Finite Perimeter

Fix an open set $\Omega \subset \mathbb{R}^n$. If $E$ is a measurable subset of $\Omega$, we may define the perimeter of $E$ in $\Omega$, denoted by $P(E;\Omega)$, to be $$P(E;\Omega) = \sup_{\...
4
votes
0answers
163 views

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 ...
3
votes
0answers
141 views

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, ...
3
votes
0answers
20 views

$||\phi-\phi_\epsilon||_{L^1(\mu)}<\epsilon$ 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 $v\mu$ ...
3
votes
0answers
53 views

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 ...
3
votes
0answers
36 views

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 ...
3
votes
0answers
114 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 ...
3
votes
0answers
45 views

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 ...
3
votes
1answer
60 views

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 $y$.
3
votes
1answer
106 views

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 $p_n\in\Pi_n$ ...
3
votes
0answers
194 views

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 ...
3
votes
0answers
108 views

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 ...
3
votes
0answers
265 views

Approximation of holomorphic functions and topological properties

So, in the last couple of lectures of my complex analysis class we've proved some approximation theorems of holomorphic functions. Eventually, we showed the following propositions: Theorem 1. Let $...
3
votes
0answers
51 views

How do I approximate $f''(x)+(E-U(x))f(x)=0$ for a piecewise $U$ and find $E$?

I am trying to approximate the solution to the equation $f''(x)+(E-U(x))f(x)=0$ where $U(x) = \begin{cases} \frac{U_0}{m}x-U_0 & \text{for $-m<x<0$} \\ \frac{-U_0}{m}x-U_0 & \text{...
3
votes
0answers
86 views

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 (...
3
votes
1answer
179 views

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 ...
3
votes
0answers
647 views

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 ...
3
votes
0answers
837 views

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 ...
2
votes
0answers
14 views

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 ...
2
votes
0answers
25 views

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 ...
2
votes
0answers
42 views

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 ...
2
votes
0answers
26 views

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 ...
2
votes
0answers
37 views

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 ...
2
votes
0answers
41 views

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 ...
2
votes
0answers
42 views

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 $\tan(x)$ ...
2
votes
0answers
79 views

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, ...
2
votes
0answers
44 views

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 ...
2
votes
0answers
170 views

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 ...
2
votes
1answer
155 views

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 ...
2
votes
0answers
60 views

Fourier series approximation of $f \in H_p^s$ rate of convergence

$H_p^s$ 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}{\...
2
votes
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 ...
2
votes
0answers
120 views

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....
2
votes
0answers
180 views

Patterns appearing in irrational approximation graphs

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 ...