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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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Chebyshev approximation for large interval

In the context of neural networks and cryptography, I would like to approximate some activation functions. However, I need to approximate them into polynomial forms for my purpose. It seems that ...
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22 views

Orthonormal basis for L2 (0,1) by using Laplacian's eigenfunctions.

A standard orthonormal basis for L2 (0,1) is given by the Fourier expansion, as described here, for example (Orthonormal Basis of $L^2$). On the other hand, it seems a standard result that the ...
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58 views

Approximate a solution for $\frac{e^{-a}(v+u)(a^x+e^{a}x\Gamma(x,a))}{\Gamma(x+1)}-u\approx 0$

Is it possible to approximate (or even find) a solution for the following equation: $$\frac{e^{-a}(v+u)(a^x+e^{a}x\Gamma(x,a))}{\Gamma(x+1)}-u\approx0,$$ where $x\ge 0$ and integer, and the ...
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20 views

Methods of approximating sine waves (given other sine waves)

I'm trying to find a way to approximate sine waves using sine waves that I already have. I have sine waves that are define by: $$f(n) = 2^{\frac{n-49}{12}}\times440$$ $$\sin(2\pi f(n)x), \sin(2\pi f(n+...
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29 views

From wavelets to curvelets

I've read that the curvelet is a generalisation of wavelet. I am now looking for references such as books, lecture note or research papers introducing the mathematical theoretic aspect of curvelet to ...
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63 views

Approximation of a sum with an integral…

Let $G$ a continuous function in $C([0,1], \mathbb R)$. I think that $$\frac{1}{N}\sum_{x \;\text{odd}\in \{1,\ldots, N\}}G\Big (\frac{x}{N}\Big )\xrightarrow{N\to +\infty}\frac{1}{2}\int_0^1G(r)dr,$$ ...
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Is there any example demonstrating nonlinearity of best polynomial approximation operator?

For any $f\in C[0,1]$, it is well known that there exists an unique $p^{*}\in P_n[0,1]$ such that $||f-p^{*}||_{\infty}=\inf\limits_{p\in P_n[0,1]}||f-p||_{\infty}$. In this fashion, one can define an ...
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How to prove the best polynomial approximation operator is continuous?

For any function $f\in C[0,1]$, it is well known that there exist an unique polynomial $p^{*}\in P_n[0,1]$ such that $||p^{*}-f||_{\infty}\leq ||p-f||_{\infty}$ for any $p\in P_n[0,1]$. In this ...
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13 views

Single hidden layer with finite #neurons limitations

I need to prove that MNN with one hidden layer, and finite number of neurons does not have compact support, i.e. the integral of the normal of f (network function) upon all R^d equal to infinity. It ...
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Approximation of Cantor function by piecewise constant function in $L^1$

Let $c(x)$ be Cantor function. How can we prove that constant $\frac{1}{2}$ gives the best approximation in $L^1$ metric? Let $ h(x)= \begin{cases} \frac14 \quad\text{for}\quad x\in[0,\frac13]\\ \...
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34 views

Problem on Riemann-Stieltjes Integration and function approximation

Let $\alpha$ be continuous and increasing function on [a,b]. Given $f$ $\in$ ${R_\alpha }$[a,b] and $\epsilon$> 0; Then, Prove that there exist (i) a step function $h$ on [a,b] with ${||h||_\infty}$$...
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an approximation to the generalized hypergeometric function

Relating to the article An approximation to the generalized hypergeometric function, I would like to calculated example of the Poisson distribution $Po(10)$. If you have the possibility, please see ...
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1answer
48 views

Extended Global Approximation Theorem

In Evans, $\textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $\partial U$ is $C^1$. Suppose as well that $u \in W^{k,p}(U)$ for some $1\leq p < \infty$. Then, there ...
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Family of graphs that have approximation ratio = 2

My question today is about the approximation algorithms. Well, for Approx-Vertex-Cover problems , we know we can get ratio of 2 just by picking an edge and taking 2 endpoints of the same and ...
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how to estimate maximum of Lebesgue function of arbitrary nodes?

Denote $S_n=\left\{x_{n}=(x_{n0},x_{n1},...,x_{nn})|a\leq x_{n0}<x_{n1}<\cdots<x_{nn}\leq b\right\}$ with $-\infty<a<b<+\infty$ and $n\geq 1$. For any $x_n\in S_n$, define the ...
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Find an L_{∞} distance between g=|x| and f=x{erf}_q(\frac{x}{\theta}).

Let $g=|x|$ and $f=x\cdot\textrm{erf}_q(\frac{x}{\theta})$, where $\textrm{erf}_q(\frac{x}{\theta})=\frac{q}{\Gamma(1/q)}\int_{0}^{\frac{x}{\theta}}\exp(-t^{q})dt$ is a generalized error function. ...
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Parametric/ homotopy version of Mergelyan theorem

Mergelyan's theorem says the following: Let $K$ be a compact subset of $\mathbb{C}$ with connected complement. Then any continuous complex-valued function on $K$ which is holomorphic in the interior ...
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1answer
28 views

Can we express the integral of the nth derivative of this function analytically?

I am currently working on an assignment with Legendre Polynomials. The integral I get stuck with is in fact the integral of the Legendre Polynomial itself i.e. $$\int \frac{1}{2^n n!} \frac{d^n}{dt^n} ...
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37 views

Why is overhat notation used both for a unit vector, $\hat{x}$, and for the closest vector, $\hat{x}$, in the best approximation theorem?

vector notation - why is overhat notation used both for a unit vector, $\hat{\mathbf x} = {\mathbf x \over || \mathbf x ||}$, and for the closest vector in a subspace $\hat{\mathbf x}$ to a vector $\...
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If graphical observations of a limit are feasible for totient approxinmation\extension onto $\mathbb R$

Essentially I want to know if the following can be considered true, despite the fact that the Euler totient is not actually a continuous function on $\mathbb R$ for which all the implications of an ...
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2answers
80 views

Does any continuous function on $[0,1]$ have a best $n$th degree polynomial approximation in the supremum norm?

Recently I am stuck in a problem in approximation theory which actually is problem in functional analysis. $C[0,1]$ is a normed vector space with $||\cdot ||_{\infty}$. $\Pi_n$ is a subspace which ...
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1answer
21 views

A density result in $W^{1,p} (\mathbb{R}^n) \cap C^{1}(\mathbb{R}^n)$

is the following result valid?: If $ u \in W^{1,p} (\mathbb{R}^n) \cap C^{1}(\mathbb{R}^n)$, then $\forall \epsilon > 0 ~ \exists f \in C_{c}^{\infty}$ s.t. $\|u-f\|_{W^{1,p}(\mathbb{R}^n)} < ...
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28 views

Showing the existence of a polynomial $p$ to approximate $f : [2,7] \rightarrow \Bbb{R}$

Let $f:[2,7] \rightarrow \Bbb{R}$ be a continuous function and for given $\epsilon >0$,we have to prove that there exists a polynomial $p$ such that $f(2)=p(2)$, $p'(2) = 0$ and $\sup\{|p(x) - f(x)|...
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54 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 ...
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22 views

How to improve derivative approximation errors along the boundary using radial basis functions

I am using radial basis functions to approximate the derivatives of a function. The test function I am using is: $g=y\cos(x)+x\sin(y)$ on the interval from 0 to $2\pi$ in both x and y directions. The ...
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1answer
29 views

Best approximation and orthogonality

Let $\mathscr{B} := \mathbb{R}^n$ equipped with the euclidian norm, let $M \in \operatorname{Mat}_{n,m}(\mathbb{R})$ a Matrix with $\operatorname{rank} M = n \le m$ and $\mathscr{A} := \{ Mx \...
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Approximation to combinations

I have to show that for very large $N$ and $n$: $$ ^{N+n_1-1}C_{n_1} \cdot ^{N+n-n_1-1}C_{n-n_1} \propto \exp{-\frac{\left(n_1-n/2 \right)^2}{\sigma ^2}}$$ where $$ \sigma ^2 = \frac{n \left(2N + ...
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1answer
54 views

How does Weierstrass' theorem follow from Mergelyan's theorem?

According to Theorems 1 and 3 in this review article we have Weierstrass: Suppose $f$ is a continuous function on a closed bounded interval $[a,b] \subset\mathbb{R}$. For every $\epsilon > 0$ ...
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1answer
113 views

Applying Bishop’s Theorem to $\langle xy;x^2y\rangle$

I am using Bishop’s Theorem in the version given by Wikipedia¹: Let $\mathfrak{A}$ be a closed subalgebra of the Banach space $C(X,ℂ)$ of continuous complex-valued functions on a compact Hausdorff ...
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Closed form of this product or approximate?

What is the closed form of this product: $$\prod_{i=1}^{k-1}\left(1-e^{-a(b- ic)^2}\right)$$ where $a,b,c$ are constants?
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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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Approximation with inequality constraints

Suppose $\mathbf x = [x_1\; x_2\; \cdots\; x_n]$ is a discrete approximation of a function at $n$ points. I want to get another approximation of this function at $n/2$ even points, say $\mathbf y = [x'...
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Comparison in the accuracy of Romberg Integration and Second Order Newton-Cotes Quadrature

Context I ask this question because I'm currently working on a program that solves the Cahn-Hilliard equation in 2D. For this project I need a subroutine to calculate the free energy functional by ...
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50 views

On a min-max approximation with polynomials

Let $n\ge 1$ be an integer. $\mathcal Q_n$ be the set of all polynomial functions over $[a,b]$, of degree exactly $n$. My question is : Is it true that $\inf_{x_0,x_1,...,x_n\in[a,b], x_0<x_1&...
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1answer
25 views

Verlet Integration to Approximate Planetary Orbit: The First Time Step

I'm currently working on a simulation of a planet orbiting binary stars, which I want to use Verlet integration to approximate. The formula is as follows: $\mathbf{p}(t_2) = 2\mathbf{p}(t_1) - \...
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47 views

Approximating $\log(1+\exp(z))$ when $z$ is complex

There exist beautiful numerical approximation for calculation of the function $$f(z) = \log(1+\exp(z)).$$ In case if $z$ is real, the following can be used $$f(z) = \begin{cases} z & z \gg 1 \\...
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Do expansion coefficients of a discontinuous function necessarily diverge?

Consider the step function on the interval $[0,1]$ with a discontinuity around 1/2 where it takes value 1/2. We know this can be expressed exactly as a Fourier series: $$\Theta(x) = \frac{4}{\pi}\sum_{...
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1answer
58 views

Numerical Analysis - Proving that the fixed point iteration method converges.

I am having some trouble with a numerical analysis proof related to the fixed point iteration method. The problem is as follows: Suppose that $f$ in $C^2[a,b]$ and for some $x$ in $(a, b)$ we have $...
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1answer
247 views

Weierstrass approximation theorem. Approximation of |x|.

I'm studying a proof of the Weierstrass approximation theorem that requires an uniform approximation using polynomials of the function |x| in the interval $[-1,1]$ i.e. we need a sequence of ...
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45 views

How to approximate a fraction of gamma functions evaluated at huge values

For sufficiently large $m$, one can approximate the function $$f:m\mapsto\frac{\Gamma \left(\frac{m+1}{2}\right)^2}{\Gamma \left(\frac{m}{2}\right) \Gamma \left(\frac{m}{2}+1\right)}$$ using the ...
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135 views

Approximating smooth function on $[0,1]$ by Bernstein polynomial (interested in approximation rate in $L^2$ norm)

Consider a smooth function $f$ on $[0,1]$ and its Bernstein polynomial of power $n$: $$B_n(f)=\sum_{k=0}^n f\left(\frac{k}{n}\right) b_{n,k}(x)$$ with $$b_{n,k}(x) = \binom{n}{k}x^k (1-x)^{n-k}.$$ ...
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1answer
77 views

Almost simple Hermite interpolation

I'm trying to use Example 4 in Section 2.5 of Philip J. Davis's book Interpolation and Approximation (Dover 1975). The aim is to fix an error in an answer I posted last night. This gives the problem a ...
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1answer
81 views

Continuity of the kernel of bounded operators under perturbation

In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator? The details: Let $(X,\| \|)$ be an infinite-dimensional real normed space. Let $A_t $ be a continuous family ...
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1answer
33 views

Find a sequence $(\phi_n)_n \subset C^{\infty}_c(\mathbb{R}^N)$ which converges in both $L^p(\nu)$ and $L^q(\mu)$ to $1_E$

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Cr}{C^{\infty}_c(\R^N)}$ Suppose we have two non-zero Borel measures on $\R^N$, labeled $\nu$ and $\mu$, and we have $1 \leq p, q < \infty$. Let $E \...
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1answer
58 views

Cramer's rule solution of the Padé approximant equations

Padé approximants are a particular type of rational approximation. The $L, M$ Padé approximant is denoted by $$[L/M] = P_L(x)/Q_M(x)$$ where $P_L(x)$ is a polynomial of degree less than or equal to $...
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13 views

Rigorous error bounds for polynomial regression

Consider a set of $N$ points $(x_i , y_i)$. I want to find a $d$ degree polynomial $P_d(x)$ that will minimize the error, $$ e_d = max_{i \in [N]} ~|P_d(x_i) - y_i| $$ The question I have is about ...
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0answers
46 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, ...
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15 views

Simultaneous asymptotic expansion in multiple points

Let $\Omega\subset\mathbb R$ be open and connected. Assume I have some non-linear, smooth function $g:\Omega \to\mathbb R$. Given disjoint base points $a_1,\ldots,a_n\in \Omega$ (and possible $\pm\...
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0answers
31 views

Boundary value problem results in system of three non-linear sine equations

I have the following equation which I am trying to find an exact solution for if possible, if not at least some approximation. The equation in general is a simple sine function, with an unknown ...
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1answer
89 views

Representation of $\pi$ using algebra and exp/log.

Can $\pi$ be represented exactly using a mixture of algebraic as well as exp/log functions, all real valued? I know it can't be done using only algebra since its transcendental, but what if we ...