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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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For any element in a Banach space, does it have a unique best approximation in a finite dimensional space?

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
14 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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25 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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48 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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17 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
27 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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17 views

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
42 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
108 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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51 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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38 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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12 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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26 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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46 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
24 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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43 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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13 views

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
39 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
188 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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43 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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2answers
119 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
68 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
70 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
32 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
56 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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0answers
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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41 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
30 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
87 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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20 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
21 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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62 views

Barycentric subdivision of planar graph approximating a (top.) path

First of all: I started reading about simplicial complexes only recently (I'll do my best to get the terminology right). For the problem: Suppose we are given a planar drawing of a cycle (i.e. Jordan ...
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2answers
104 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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0answers
82 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
44 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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38 views

proof of chebychev polynomial

Can someone explain the Intuition or the reasoning of the following Chebychev Theorem (page 72 Theorem 2.7.2): In order that the ordinary polynomial $P(x)$ among all polynomials of degree < $n$ ...
2
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1answer
54 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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0answers
27 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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38 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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1answer
46 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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0answers
89 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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0answers
29 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 ...
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0answers
35 views

Chebyshev coefficients of $e^{-x}$

Im trying to derive, given $n$, the Chebyshev coefficients $c_k$ of $e^{-x}$ on [-1,1]. That is $c_k=\frac{(e^{-x},T_k)_w}{(T_k,T_k)_w}$, $k=0,1,...,n.$ I have problems computing $(e^{-x},T_k)_w= \...
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0answers
23 views

Spline Approximation Results in $L^2(\mathbb{R})$ norm

I have seen in several papers that one can approximate a function $f \in C^{(k-1)}$ via splines in $S_\pi^k$ of order $k$ with extended knot sequence $\pi$ using a local approximation operator $Q: C^{(...
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1answer
43 views

approximating irrational roots of algebraic equations with the Pierce expansion

Let $ p (x) = 1 - x \lfloor \frac{1}{x} \rfloor $ then the Pierce expansion of a real number $x \in {R}$ is expressed by \begin{equation} x_1 = \sum_{n = 1}^{\infty} (- 1)^{n + 1} \prod_{m = 1}^n ...
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2answers
27 views

Approximating a piece-wise function

I would like to approximate a piece-wise function. The aim is to get a function as $f(x) \approx ...$ without piece-wise definition (only one expression, not depending of $x \leq 1$ or $x \geq 1$), ...
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1answer
14 views

Question about $O$

Consider two non-negative sequences $f(n)$ and $g(n)$ and suppose that $\exists C>0, \ \forall \varepsilon >0, \exists N \in \mathbb{N}$ s.t. $$\forall n \geq N, \ \ f(n)<C \cdot g(n)+n^{\...
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2answers
120 views

What is the optimal Fourier series convergence rate estimate for $|x|$?

What is the known best estimate of the rate of convergence in $\|\cdot\|_\infty$ (or maximal absolute value) of the Fourier series of $|x|,\, x\in[-1,1]$? If I look at the coefficients of the Fourier ...
3
votes
1answer
84 views

Smooth floor function

I want a monotonic function on the positive real numbers that behaves like floor but in smooth way, like smoothstep but for all integers. It should follow this simple rule. slope is zero at ...