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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69 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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54 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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43 views

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
157 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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1answer
66 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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8 views

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
242 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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2k views

Is Ramanujan's approximation for the factorial optimal, or can it be tweaked? (answer below)

Ramanujan's famous factorial approximation, $$n!\approx\sqrt{\pi}\left(\frac{n}{e}\right)^n\root\LARGE{6}\of{8n^3+4n^2+n+\frac{1}{30}}$$ is far more accurate that the Stirling approximation when ...
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1answer
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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1answer
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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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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1answer
175 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 ...
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673 views

best approximation of $\sqrt{2}$

The approximation \begin{align} \sqrt{2} &\approx \frac{1}{8} \operatorname{csch}\left(\frac{3\pi}{2}\right) \operatorname{sech}^3(\pi) \, \left[2+3 \, \sinh\left(\frac{\pi}{2}\right)-\sinh\left(\...
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1answer
97 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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1answer
600 views

Magnitude of n-th factorial

In connection with a riddle on The Riddler, I would like to know how to evaluate even crudely the order of magnitude of an iterated factorial like $$(\ldots(9\underbrace{!)!\ldots)!}_{n\text{ ...
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1answer
145 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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1answer
97 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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68 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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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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45 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 $\tan(x)$ ...
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25 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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89 views

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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1answer
76 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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65 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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1answer
267 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
374 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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263 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
84 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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3answers
261 views

Possible Pythagorean relation with Golden Ratio $ \phi^2+e^2 \approx \pi^2$

While study Numerics and playing with famous constants ($e$, $\pi$, Golden ratio) I came across the following relation $$ \color{blue}{1.6^2+2.7^2 = 9.85\approx 3.14^2}$$ This is nothing special but ...
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1answer
130 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
38 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
161 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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2answers
60 views

Sobolev approximation lifts to $L^p$ convergence of the exterior powers

I am reading the book "Geometric Function Theory and Non-linear Analysis", where the following claim is used: Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set. Let $f \in W^{1,s}(\Omega,\...
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0answers
84 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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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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35 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
93 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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74 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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1answer
32 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 ...
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115 views

Can we provide a good estimation for $(n!)!$?

I was thinking about this $$(n!)!$$ for $n\in\mathbb{N}$. I wanted to find a suitable approximation, or in any case a very good estimation for this. My first idea was to use Stirling ...
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123 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 ...
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1answer
250 views

$\rm Proximinal$ and $\rm \varepsilon-proximinal$ set

When $K$ is a nonempty subset of a metric space $M$, $\forall x \in M$ let $P_K(x)=\{y \in K: d(x,y)=d_K(x)=\inf_{k \in K}d(x,k)\}$ (metric projection) $\bullet$ The set $K$ is $proximinal$ in $M$ ...
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1answer
436 views

Approximation of a continuous function by piecewise constant function

Let $f:(0,1) \to \mathbb{R}$ be continuous and increasing. Define $$f_n(t) := \sum_{i=0}^{n-1} f(T^n_i)\chi_{(T^n_i, T^n_{i+1})}(t)$$ where $\{T^n_0, T^n_1, ..., T^n_n\}$ is a uniform partition of $(0,...
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1answer
196 views

Advantage of Bernstein polynomial basis

The well-known "Bernstein polynomials" on the interval [0,1] are defined as $$ B_{N,i}(x)=\binom{n}{i}x^{i}(1-x)^{n-i}, \ \ i=0,...,N. $$ My question is about advantage of these polynomials in ...
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67 views

Approximating Log(Gamma(z)) for small z as Log(Gamma(z + 1)) - Log(z)

I'd like to implement a numerical approximation to the log Gamma function, and I found Gergő Nemes' approximation described here: https://en.wikipedia.org/wiki/Stirling%27s_approximation. This seems ...
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55 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 ...
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2answers
320 views

Stirling's Approximation Proof

I was searching for the reason why Stirling's Approximation holds true. I found the website Stirling's Approximation which apparently shows why this is the case. Does this part of the equation make ...
2
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1answer
56 views

Proving this condition for convergence in a Banach space

I have difficulty proving the following claim from a paper (a free version is here, see Lemma 2.4 on page 9): Let in a Banach space $X$ a sequence $\{x_n\}_{n=1}^\infty$ be given. Assume that for ...
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270 views

Simpson's Rule in Matlab [closed]

I have made the following code based on Simpson's expansion: function I = simprule(f, a, b, n) h = (b-a) / n; x = a:h:b; S = 0; L = 0; for l = 1:2:n %generates the odd number array S = S + 4*...

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