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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104 views

Error analysis of approximating Fourier transforms

Consider the problem of computing the Fourier transform of a function, $f(x).$ $$ \hat{f}(k) = \int_{-\infty}^{\infty} dx~ f(x)~ e^{i k x} .$$ Suppose I want to approximate this transform by a ...
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159 views

Approximation of conditional expectation of unknown function

I am given a multidimensional markovian stochastic process $X_1,X_2,...X_n$ with continuous state space and unknown to me function $V$. I want to approximate expectation $E(V(X_k)|X_{k-1} = x)$ ...
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67 views

Relating the Nonlinear and Linear operators in the Homotopy Analysis Method

The question refers to chapter two of the book Liao, Shijun. Homotopy analysis method in nonlinear differential equations. Beijing: Higher Education Press, 2012. Link to book pdf from Chinese .edu ...
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22 views

Approximation in probability

To prove a CLT, I need first to prove that the following approximation holds $$ \sqrt{n}\sum_{j=1}^n\left(\left|\int_{(j-1)/n}^{j/n}\nu_s\,dW_s\right|\,\left|\int_{j/n}^{(j+1)/n}\nu_s\,dW_s\right|-\...
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48 views

A strengthening of Stone–Weierstrass which applies to arbitrary closed intervals

Note in this question, we concern ourselves only with the space $C([a,b])$ of continuous real-valued functions on compact intervals $[a,b]$. The Müntz–Szász theorem is a well-known result related to ...
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32 views

Taylor Series Expansion on error propagation.

I am reading through Stoer and Bulirsch's Introduction to Numerical Analysis. In their section on error propagation they are describing a derivation of the Jacobian as it related to a problems ...
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1answer
51 views

Clarification of assumptions made in deriving error of implicit midpoint rule

In my derivation for $y^\prime = f(t,y)$, I begin by writing the method as an expression which should simplify to the error, by substitution of the exact solution \begin{equation} y(t_{n+1}) - y(t_n) -...
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81 views

Computing the solution of an ODE using power series

I have a system of ODEs defined on $\mathbb{R}\times\mathbb{S}$, $$\begin{aligned}\dot{x}={ }&y\\\dot{y}={ }&-0.2y+\frac{300\cos(2)\sin(x)}{1.8(1.3+\cos(x-2))-2\sin(x)\sin(2)},\end{aligned}$$ ...
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59 views

Polynomial approximation of a function in a chosen interval

I don't know if this should be in the math forum or the computer science one, but I think it is more relevant in the math section. I would like to approximate nonlinear functions typically used in ...
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76 views

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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158 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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59 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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33 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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188 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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32 views

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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65 views

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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52 views

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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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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70 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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132 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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57 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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195 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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28 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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40 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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69 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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85 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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95 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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143 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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61 views

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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24 views

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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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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65 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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64 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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234 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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347 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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52 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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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
124 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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147 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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37 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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81 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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25 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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82 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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34 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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90 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 ...
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62 views

randomized approximate matrix inverse or adjoint of a square matrix

I have been reading about some random matrix theory, JL, and related topics and am wondering if there are any methods to calculate an approximate inverse of a SPD matrix $\mathbf{A}$, or possibly even ...
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30 views

Closed linear span of translations of simple step functions

This paper utilizes Wiener's tauberian theorem to indicate that the closed linear span of translations of any simple step function is equal to $L^p[a,b]$, where $1< p \leq \infty$ and $[a,b]$ are ...